arXiv:1609.09153v2 [math.NT] 26 Jan 2017 · Another relevant work is the paper of Louboutin, Park,...

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ON Dℓ-EXTENSIONS OF ODD PRIME DEGREE ℓ

HENRI COHEN AND FRANK THORNE

Abstract. Generalizing the work of A. Morra and the authors, we give explicit formulas for the Dirichletseries generating function of Dℓ-extensions of odd prime degree ℓ with given quadratic resolvent. Over thecourse of our proof, we explain connections between our formulas and the Ankeny-Artin-Chowla conjecture,the Ohno–Nakagawa relation for binary cubic forms, and other topics. We also obtain improved upper boundsfor the number of such extensions (over Q) of bounded discriminant.

1. Introduction

The theory of cubic number fields is, in many respects, well understood. One reason for this is thatthe Delone-Faddeev [20] and Davenport-Heilbronn [19] correspondences parametrize cubic fields in termsof binary cubic forms, up to equivalence by an action of GL2(Z), and satisfying certain local conditions.Therefore questions about counting cubic fields can be reduced to questions about counting lattice points,and this idea has led to asymptotic density theorems as well as other interesting results.

In more recent work, Bhargava [5, 6] obtained similar parametrization and counting results for S4-quarticand S5-quintic fields. However, generalizing this work to number fields of arbitrary degree ℓ seems difficult, ifnot impossible: the parametrizations of S3-cubic, S4-quartic, and S5-quintic fields are all by prehomogeneousvector spaces, and for higher degree fields there is no apparent prehomogeneous vector space for which onecould hope to establish a parametrization theorem.

In [14] and [16], A. Morra and the authors contributed to the cubic theory by giving explicit formulas forthe Dirichlet generating series of discriminants of cubic fields having given resolvent. For example, we havethe explicit formula

(1.1) Φ−107(s) :=∑

[K:Q]=3Disc(K)=−107n2

n−s = −1

2+

1

2

(1 +

2

32s

) ∏

(321p )=1

(1 +

2

ps

)+

(1 +

2

32s

)∏

p

(1 +

ω(p)

ps

),

where ω(p) is equal to 2 or −1 if p is totally split or inert in the unique cubic field of discriminant 321,determined by the polynomial x3 − x2 − 4x+ 1, and ω(p) = 0 otherwise. Similar formulas hold when −107is replaced by any other fundamental discriminant D; the formula has one main term, and one additionalEuler product for each cubic field of discriminant −D/3, −3D, and −27D.

The proofs involve class field theory and Kummer theory; see also work of Bhargava and Shnidman [7]obtaining related results through a study of binary cubic forms.

The object of the present paper is to generalize the theory developed in [14] and [16] to degree ℓ extensionshaving Galois group Dℓ, for any odd prime ℓ. (See also [10, 11, 13, 33, 44], among other references, for furtherrelated results.)

Date: November 7, 2018.

1

http://arxiv.org/abs/1609.09153v2

2 HENRI COHEN AND FRANK THORNE

Let L/k be an extension1 of odd prime degree ℓ, let N = L be a Galois closure of L, and assume thatGal(N/k) ≃ Dℓ, the dihedral group with 2ℓ elements. We will refer to any such L as a Dℓ-extension of k, ora Dℓ-field when k = Q. Below we also refer to Fℓ-extensions with the analogous meaning.

There exists a unique quadratic subextension K/k of N/k, called the quadratic resolvent of L, withGal(N/K) ≃ Cℓ, and a nontrivial theorem of J. Martinet involving the computation of higher ramificationgroups (see Propositions 10.1.25 and 10.1.28 of [8]) tells us that its conductor f(N/K) is of the formf(N/K) = f(L)ZK , where f(L) is an ideal of the base field k, and that the relative discriminant d(L/k) of

L/k is given by the formula d(L/k) = d(K/k)(ℓ−1)/2f(L)ℓ−1.We study the set Fℓ(K) of Dℓ-extensions of k whose quadratic resolvent field is isomorphic to K. (Here

and in the sequel, extensions are always considered up to k-isomorphism.) More precisely, we want tocompute as explicitly as possible the Dirichlet series2

Φℓ(K, s) =1

ℓ− 1+

∑

L∈Fℓ(K)

1

N (f(L))s,

where N (f(L)) = Nk/Q(f(L)) is the absolute norm of the ideal f(L).

Our most general result is Theorem 6.1, which we specialize to a more explicit version in the case k = Qas Theorem 7.3. This should be considered as the most important result of this paper. In Section 9 we provethat our formulas can always be brought into a form similar to (1.1). For example, we have

Φ5(Q(√5), s) =

1

20

(1 +

4

5s

) ∏

p≡1 (mod 5)

(1 +

4

ps

)+

1

5

(1− 1

5s

) ∏

p≡1 (mod 5)

(1 +

ωE(p)

ps

),

where E is the field defined by x5 + 5x3 + 5x− 1 = 0 of discriminant 57, and ωE(p) = −1, 4, or 0 accordingto whether p is inert, totally split, or other in E.

In a companion paper, joint with Rubinstein-Salzedo [15], we investigate a curious twist to this story.Taking the n = 1 term of formula (1.1) (or, rather, its generalization to any D) yields the nontrivial identity

(1.2) N3(D∗) +N3(−27D) =

{N3(D) if D < 0,

3N3(D) + 1 if D > 0,

for any fundamental discriminant D, where D∗ = −3D if 3 ∤ D and D∗ = −D/3 if 3 | D. (Here N3(k) is thenumber of cubic fields of discriminant k. Note that there are no cubic fields of discriminant −3D if 3 | D.)This identity was previously conjectured by Ohno [38] and then proved by Nakagawa [36], as a consequenceof an ‘extra functional equation’ for the Shintani zeta function associated to the lattice of binary cubic forms.Our generalization of (1.1) thus subsumes the Ohno–Nakagawa theorem (1.2).

Our proof there used the Ohno–Nakagawa theorem, but in [15] we further develop some of the techniquesof this paper (in particular, of Section 8) to give another proof of (1.2) and give a generalization to any primeℓ ≥ 3. For ℓ > 3 our work relates counts of Dℓ-fields (the right-hand side of (1.2)) to counts of Fℓ-fields ℓ(the left-hand side), where Fℓ is the Frobenius group of order ℓ(ℓ− 1), whose definition is recalled in Section9. (Note that S3 = D3 = F3.) The result involves a technical (Galois theoretic) condition on the Fℓ-fieldswhich is not automatically satisfied for ℓ > 3, and we defer to [15] for a complete statement of the results. It

1A remark on our choice of notation: Readers familiar with [16] or [14] should note that by and large we adopt the notationof [16] and the progression of [14]; the reader knowledgeable with the latter paper can immediately see the similarities anddifferences. (See also Morra’s thesis [34] for a version of [14] with more detailed proofs.) What was called (K2,K, L,K′

2) in [14]will now be called (K,L,Kz,K

′) (so that the main number field in which most computations take place is Kz), and the fieldnames (k, kz, N,Nz) stay unchanged. The primitive cube root of unity ρ is replaced by a primitive ℓth root of unity ζℓ.

2The series also depends on the base field k, which we do not include explicitly in the notation.

ON Dℓ-EXTENSIONS OF ODD PRIME DEGREE ℓ 3

is however important to note that, as for the cubic case, even the case n = 1 of our Dirichlet series identitiesgives interesting results: for instance, for any negative fundamental discriminant −D coprime to 5, we have

(1.3) NF5

((−1)053D2

)+NF5

((−1)055D2

)+NF5

((−1)057D2

)= ND5

((−D)2

)+ND5

((−5D)2

),

and if instead D > 1, then we have

(1.4) NF5

((−1)253D2

)+NF5

((−1)255D2

)+NF5

((−1)257D2

)= 5

(ND5

(D2

)+ND5

((5D)2

))+ 2 .

(Here, NG((−1)rX) denotes the number of quintic fields with discriminant exactly equal to X, with r pairsof complex embeddings, and whose Galois closure has Galois group G over Q.)

If we want an identity counting D5-fields of discriminant (±D)2 or (±5D)2 alone, then the left side of (1.3)and (1.4) becomes more complicated, and involves the Galois condition mentioned above. The relevance tothe present paper is that it is precisely those Fℓ-fields counted by this identity that yield Euler products. Wedescribe this in more detail in Section 9.

There is one further curiosity that emerges in our work: a connection to a well-known conjecture attributedto3 Ankeny, Artin, and Chowla [1] which states that if ℓ ≡ 1 (mod 4) is prime and ǫ = (a + b

√ℓ)/2 is the

fundamental unit of Q(√ℓ), then ℓ ∤ b. As we will see, the truth or falsity of the conjecture will be reflected in

our explicit formula for Dℓ-extensions having quadratic resolvent Q(√ℓ). Note that the conjecture is known

to be true for ℓ < 2 ·1011 (see [40]), but on heuristic grounds it should be false: if we assume independence ofthe divisibility by ℓ, the number of counterexamples for ℓ ≤ X should be around log(log(X))/2; in addition,numerous counterexamples can easily be found for “fake” quadratic fields, see e.g., [17, 37].

Separately, we obtain improved bounds for the number Nℓ(Dℓ,X) of Dℓ-fields L with |Disc(L)| < X:

Theorem 1.1. We have

(1.5) Nℓ(Dℓ,X) ≪ℓ,ǫ X3

ℓ−1− 1

ℓ(ℓ−1)+ǫ.

This improves on Kluners’s [29] bound of O(X3

ℓ−1+ǫ). The proof (in Section 10) is independent of the rest

of the paper, and is an immediate consquence of applying Ellenberg, Pierce, and Wood’s [22] recent boundson ℓ-torsion in quadratic fields within Kluners’s method.

Our work follows several other papers studying dihedral field extensions. Much of the theory (such asMartinet’s theorem) is described in the first author’s book [8]. Another reference is Jensen and Yui [26],who studied Dℓ-extensions from multiple points of view. They proved that if ℓ ≡ 1 (mod 4) is a regularprime, then no Dℓ-extension of Q has discriminant a power of ℓ, and we will recover and strengthen theirresult. Jensen and Yui also studied the problem of constructing Dℓ-extensions, and gave several examples.

Another relevant work is the paper of Louboutin, Park, and Lefeuvre [32], who developed a general classfield theory method to construct real Dℓ-extensions. These problems have also been addressed in the functionfield setting by Weir, Scheidler, and Howe [43].

Since some of the proofs are quite technical, we give a detailed overview of the contents of this paper.We begin in Section 2 with a characterization of the fields L ∈ Fℓ(K) using Galois and Kummer theory.

These fields are in bijection with elements of Kz := K(ζℓ) modulo ℓth powers, satisfying certain restrictionswhich guarantee that the associated Kummer extensions of Kz descend to degree ℓ extensions of k. Writing

such an extension as Kz( ℓ√α) with αZKz =

∏0≤i≤ℓ−2 a

gi

i qℓ, we further characterize these fields in terms ofconditions on the ai and an associated member u of a Selmer group associated to Kz.

3Ankeny, Artin, and Chowla did not conjecture this in [1], although they did explicitly ask if it is true. Mordell [35] attributedthe conjecture to them in followup work, where he proved the conjecture for regular primes.


These conditions are described in terms of the group ring Fℓ[G], where G = Gal(Kz/k). Groups such as

K∗z/K

∗zℓ, Cl(Kz)[ℓ], and the Selmer group are naturally Fℓ[G]-modules, and our conditions correspond to

being annihilated by certain elements of Fℓ[G] (see Definition 2.2).In Section 2 we also study the subfields of Kz/k, with particular attention to a degree ℓ−1 extension K ′/k

called the mirror field of K; we will see that much of the arithmetic of prime splitting in various extensionscan be conveniently expressed in terms of K ′.

The reader who is willing to take our technical computations for granted is advised to look only at thenecessary definitions in the intermediate sections and to skip directly to Section 6.

In Section 3, we give an expression for the ‘conductor’ f(L) in terms of the quantities ai and u defined inSection 2. The main result, Theorem 3.8, was proved by the first author, Diaz y Diaz, and Olivier in [10] intheir study of cyclic extensions of degree ℓ. Unfortunately the results of that section are rather complicatedto state, and oblige us to introduce a fair amount of notation.

In Section 4 we begin to study the fundamental Dirichlet series using the results proved in Section 3, andin Section 5 we study the size of a certain Selmer group appearing in our formulas. The latter section isheavily algebraic and again appeals heavily to the results of [10].

In Section 6, we put everything together to obtain our most general formula (Theorem 6.1) for Φℓ(K, s), ageneralization of the main theorem of [14]. Subsequently we work to make everything more explicit, for themost part specialized to the case k = Q. In Section 7 we compute various quantities appearing in Theorem6.1 for k = Q, leading to Theorem 7.3, a more explicit specialized version of Theorem 6.1. We also obtainasymptotics for counting Dℓ-extensions of Q, proved in Corollary 7.5.

The formula of Theorem 7.3 involves a somewhat complicated group Gb, and in Section 8 we further studyits size. The main result is the Kummer pairing of Theorem 8.2, familiar from (for example) the proof ofthe Scholz reflection principle, and fairly simple to prove. One important input (Proposition 8.1) is a verynice relationship, due essentially to Kummer and Hecke, between the conductor of Kummer extensions ofKz, and congruence properties of the ℓth roots used to generate them.

In Section 8 we also explore the connection to the conjecture of Ankeny, Artin, and Chowla mentionedabove. The truth or falsity of this conjecture will then be reflected in our explicit formula (Proposition

9.2) for Φℓ(Q(√ℓ), s), and in Corollary 9.3 we will give a proof of an observation of Lemmermeyer, that the

existence of Dℓ-fields ramified only at ℓ is equivalent to the falsity of the Ankeny-Artin-Chowla conjecture.In Section 9 we further study the characters of the group Gb, and prove (in Theorem 9.1) that each such

character corresponds to an Fℓ-extension E/k, such that the values of χ correspond to the splitting typesof primes in E. This was done for ℓ = 3 and k = Q in [16], but in Theorem 9.1 we do not require k = Q.

It is here that the connection to the Ohno–Nakagawa theorem emerges; for ℓ = 3 and k = Q, we establishedin [16] (using Ohno–Nakagawa) that the set of characters of Gb corresponds precisely to a suitable and easilydescribed set of fields E. For ℓ > 3 we require the generalization of Ohno–Nakagawa established in [15], andso in Section 9 we say a bit more about the results of [15] and explain their relevance. We also prove an

explicit formula valid for the ‘special case’ K = Q(√ℓ).

Acknowledgements

We would like to thank Michael Filaseta, David Harvey, Franz Lemmermeyer, Hendrik Lenstra, DavidRoberts, and John Voight for helpful comments and suggestions.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1201330 and by the National Security Agency under a Young Investigator Grant.


2. Galois and Kummer Theory

2.1. Galois and Kummer theory, and the Group Ring. We will use the results of [10], but beforestating them we need some notation. We denote by ζℓ a primitive ℓth root of unity, we set Kz = K(ζℓ),kz = k(ζℓ), Nz = N(ζℓ), and we denote by τ , τ2, and σ generators of kz/k, K/k, and N/K respectively, withτ ℓ−1 = τ22 = σℓ = 1.

The number ζℓ could belong to k, or to K, or generate a nontrivial extension of K of degree dividingℓ−1. These essentially correspond respectively to cases (3), (4), and (5) of [14] (cases (1) and (2) correspondto cyclic extensions of k of degree ℓ, which have been treated in [10]). Cases (3) and (4) are considerablysimpler since we do not have to adjoin ζℓ to K to apply Kummer theory.

We are particularly interested in the case k = Q, in which case either [Kz : K] = ℓ − 1, or [Kz : K] =

(ℓ − 1)/2, i.e., K ⊂ kz, which is equivalent to K = Q(√ℓ∗) with ℓ∗ = (−1)(ℓ−1)/2ℓ. To balance generality

and simplicity, we assume that k is any number field for which [kz : k] = ℓ − 1. Then, as for k = Q thereare two possible cases: either [Kz : K] = ℓ − 1, which we call the general case, or K ⊂ kz = Kz and[Kz : K] = (ℓ − 1)/2, which we will call the special case. Note that if ℓ = 3 this means that ζℓ ∈ K, so weare in case (4), but there is no reason to treat this case separately. It should not be particularly difficult toextend our results to any base field k, as was done in [10].

We set the following notation:

• We let g be a primitive root modulo ℓ, and also denote by g its image in F∗ℓ = (Z/ℓZ)∗.

• We let G = Gal(Kz/k). Thus in the general case G ≃ (Z/2Z) × (Z/ℓZ)∗, while in the special caseG = Gal(kz/k) ≃ (Z/ℓZ)∗. We denote by τ the unique element of Gal(kz/k) such that τ(ζℓ) = ζgℓ ,so that τ generates Gal(kz/k), and we again denote by τ its lift to Kz or Nz.

The composite extension Nz = NKz is Galois over k, and σ and τ naturally lift to Nz. In the generalcase, τ and σ commute; in the special case, τ2 is a generator of Gal(Kz/K) and τ2 can be taken to be anyodd power of τ , for instance τ itself, so that τστ−1 = σ−1.

This information is summarized in the two Hasse diagrams below, depicting the general and special casesrespectively.


Nz

<τ>

❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧

<σ>

Nz

<τ2>

tttttttttttttttttttttt

<σ>

N

<τ2>

✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞

<σ>≃Cℓ

N

<τ>

✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞

<σ>≃Cℓ

Kz ⊇ pz

<τ>

♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥

<τ2>

⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤

Kz = kz ⊇ pz

<τ2>

✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈

<τ>

⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤

ℓL

ℓ

ℓL

ℓ

K ⊇ p

<τ2>

✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆

kz = k(ζℓ) ⊇ pk

<τ>

♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦

K ⊇ p

<τ>

✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆

k ⊇ p k ⊇ p

In the above p, p, pk , pz indicate our typical notation (to be used later) for primes of k,K, kz ,Kz respec-tively. We write ℓL to indicate the existence of ℓ fields isomorphic to L.

Lemma 2.1. For a mod (ℓ− 1) and b mod 2, set

ea =1

ℓ− 1

∑

j mod (ℓ−1)

gajτ−j ∈ Fℓ[G] and, in the general case, e2,b =1

2

∑

j mod 2

(−1)bjτ−j2 .

The ea form a complete set of orthogonal idempotents in Fℓ[G], as do the e2,b in the general case, so in thegeneral case any Fℓ[G]-module M has a canonical decomposition M =

∑a mod (ℓ−1), b mod 2 eae2,bM , while in

the special case we simply have M =∑

a mod (ℓ−1) eaM .

Proof. Immediate and classical; see, e.g., Section 7.3 of [21]. �

We set the following definitions:

Definition 2.2. In the group ring Fℓ[G], we set

T =

{{τ2 + 1, τ − g} in the general case ,

{τ + g} in the special case .

(1) We define ι(τ2 + 1) = e2,1 = 12(1 − τ2), and for any a we define ι(τ − ga) = ea, so that for instance

ι(τ + g) = e(ℓ+1)/2.(2) For any Fℓ[G] module M , we denote by M [T ] the subgroup annihilated by all the elements of T .

Lemma 2.3. Let M be an Fℓ[G]-module.

(1) For any t ∈ T we have t ◦ ι(t) = ι(t) ◦ t = 0, where the action of t and ι(t) is on M .(2) For all t ∈ T we have M [t] = ι(t)M and M [ι(t)] = t(M).


(3) If x ∈ M [t] then ι(t)(x) = x.

Proof. This follows from Lemma 2.1. In particular, τea = gaea, so that the image of τ−ga is∑

b6=a eaM . �

2.2. The Bijections.

Proposition 2.4. (1) There exists a bijection between elements L ∈ Fℓ(K) and classes of elements

α ∈ (K∗z /K

∗zℓ)[T ] such that α 6= 1, modulo the equivalence relation identifying α with αj for all j

with 1 ≤ j ≤ ℓ− 1.(2) If α ∈ K∗

z is some representative of α, the extension L/k corresponding to α is the field Kz( ℓ√α)G,

i.e., the fixed field of Kz( ℓ√α) by a lift of G = Gal(Kz/k) to Gal(Kz( ℓ

√α)/k).

Proof. First assume that L ∈ Fℓ(K); since ζℓ ∈ Kz, by Kummer theory cyclic extensions of degree ℓ of Kz

are of the form Nz = Kz( ℓ√α), where α 6= 1 is unique in K∗

z /K∗zℓ modulo the equivalence relation mentioned

in the proposition. As Nz determines L up to conjugacy, we must prove that α is annihilated by T .Writing Nz = Kz(θ) with θℓ = α, we may assume the generator σ chosen so that σ(θ) = ζℓθ. Set ε = 1 if

we are in the general case, ε = −1 if we are in the special case, so that τστ−1 = σε. We have τ(ζℓ) = ζgℓ , sothat

σ(τ(θ)) = τ(σε(θ)) = τ(ζεℓ )τ(θ) = ζεgℓ τ(θ) ,

hence if we set η = τ(θ)/θεg we have σ(η) = ζεgℓ τ(θ)/ζεgℓ θεg = η. It follows by Galois theory that η ∈ Kz, so

that τ(α)/αεg = ηℓ ∈ K∗zℓ, hence that α ∈ (K∗

z /K∗zℓ)[τ − εg].

Concerning τ2 (in the general case only), the relation τ2στ−12 = σ−1 similarly shows that σ(θτ2(θ)) = θτ2(θ)

so that α ∈ (K∗z /K

∗zℓ)[τ2 + 1], in other words α ∈ (K∗

z/K∗zℓ)[T ] as desired.

To conclude, we must prove that each α ∈ (K∗z/K

∗zℓ)[T ] determines such an L ∈ Fℓ(K). Again write

θ = ℓ√α with σ(θ) = ζℓθ and Nz = Kz(θ). Define an automorphism τ of Nz, agreeing with τ on Kz, by

writing τ(θ) = ηθεg (ε = ±1 as before), where ηℓ = τ(α)/αεg ∈ Kℓz, so that η ∈ Kz is well-defined up to an

ℓth root of unity, and we make an arbitrary such choice.Computations show that τσε(θ) = στ(θ) and that τ ℓ−1(θ) is θ times a root of unity. Each τσi is also a

lift of τ .In the general case, we check that there is a unique such lift, which we denote simply by τ , for which

τ ℓ−1(θ) = θ. Write τ2(θ) = η2/θ with ηℓ2 = ατ2(α) where η2 is in Kz and indeed kz. We check that τ22(θ) = θ

and τ2σ(θ) = σ−1τ2, so that by rewriting τ2 as τ2 we see that Nz/k is Galois with Galois group Cℓ−1×Dℓ, asrequired. Here the choice of lift τ2 is not uniquely determined: Dℓ has ℓ elements of order 2, correspondingto the ℓ conjugate subextensions L/k of degree ℓ.

In the special case, rewriting τ as τ we now have τ ℓ−1 = 1 regardless of the choice of lift: we haveτ ℓ−1(θ) = ζ iℓθ for some integer i, so that unless i ≡ 0 (mod ℓ), τ is of order ℓ(ℓ− 1). We already know thatNz/k is Galois, as the τ rσs are distinct automorphisms of Nz/k for 0 ≤ τ < ℓ − 1, 0 ≤ σ < ℓ. We havealready proved that Gal(Nz/k) is nonabelian, and in particular noncyclic, hence i = 0. So τ ℓ−1 = 1 andGal(Nz/k) has the required presentation.

�

Recall from [8] the following definition:

Definition 2.5. We denote by Vℓ(Kz) the group of (ℓ-)virtual units of Kz, in other words the group ofu ∈ K∗

z such that uZKz = qℓ for some ideal q of Kz, or equivalently such that ℓ | vpz(u) for any prime ideal

pz of Kz. We define the (ℓ-)Selmer group Sℓ(Kz) of Kz by Sℓ(Kz) = Vℓ(Kz)/K∗zℓ.

The following lemma shows in particular that the Selmer group is finite.


Lemma 2.6. We have a split exact sequence of Fℓ[G]-modules

1 −→ U(Kz)

U(Kz)ℓ−→ Sℓ(Kz) −→ Cl(Kz)[ℓ] −→ 1 ,

where the last nontrivial map sends u to the ideal class of q such that uZKz = qℓ.

Proof. Exactness follows from the definitions, and the sequence splits because ℓ ∤ |G| (see for example [12,Lemma 3.1] for a proof). �

From Lemma 2.3 we extract the following technical result.

Lemma 2.7. Given t ∈ T and α ∈ K∗z such that t(α) is a virtual unit, we have t(α) = γℓt(u) for some

γ ∈ K∗z and some virtual unit u.

Moreover, if α is annihilated modulo K∗zℓ by t′ 6= t ∈ T , we may choose u to be annihilated by t′ in Sℓ(Kz).

Proof. Given t and α, (1) of Lemma 2.3 applied to M = K∗z/K

∗zℓ implies that ι(t)(t(α)) ∈ K∗

zℓ. Since

t(α) is a virtual unit, its image t(α) is annihilated by ι(t) in the Selmer group. By Lemma 2.3 applied to

M = Sℓ(Kz), we have t(α) = t(β) for some β ∈ Sℓ(Kz), giving the first result. For the second, we replaceeach of the modules M by M [t′]: since t and t′ commute, if α ∈ M is annhilated by t′, so is t(α). �

Proposition 2.8. (1) There exists a bijection between elements L ∈ Fℓ(K) and equivalence classes ofℓ-tuples (a0, . . . , aℓ−2, u) modulo the equivalence relation

(a0, . . . , aℓ−2, u) ∼ (a−i, . . . , aℓ−2−i, ugi)

for all i (with the indices of the ideals a considered modulo ℓ− 1), where the ai and u are as follows:

(a) The ai are coprime integral squarefree ideals of Kz such that if we set a =∏

0≤i≤ℓ−2 agi

i then the

ideal class of a belongs to Cl(Kz)ℓ, and a ∈ (I(Kz)/I(Kz)

ℓ)[T ], where as usual I(Kz) denotesthe group of (nonzero) fractional ideals of Kz.

(b) u ∈ Sℓ(Kz)[T ], and in addition u 6= 1 when ai = ZKz for all i.(2) Given (a0, . . . , aℓ−2), a, and u as in (a), the field L ∈ Fℓ(K) is determined as follows: There exist

an ideal q0 and an element α0 ∈ Kz such that aqℓ0 = α0ZKz with α0 ∈ (K∗z /K

∗zℓ)[T ]. Then L is any

of the ℓ conjugate degree ℓ subextensions of Nz = Kz( ℓ√α0u), where u is an arbitrary lift of u.

Proof. Given L, associate Nz = Kz( ℓ√α) as in Proposition 2.4. We may write uniquely αZKz =

∏0≤i≤ℓ−2 a

gi

i qℓ,

where the ai are coprime integral squarefree ideals of Kz, and they must satisfy the conditions of (a).

Each a which thus occurs satisfies aqℓ = α0ZKz for some α0 with α0 ∈ (K∗z /K

∗zℓ)[T ], and for each a we

arbitrarily associate such an α0. Given aqℓ = αZKz , u := α/α0 is a virtual unit; writing u for its class inSℓ(Kz), u is annhiliated by T because both α and α0 are.

This establishes the bijection, and we conclude by observing the following:

• The elements α and β give equivalent extensions if and only if β = αgiγℓ for some element γ and

some i modulo ℓ − 1, and then if α0ZKz =∏

j agj

j qℓ and α = α0u, we have on the one hand

βZKz =∏

j agj

j−iqℓ1 for some ideal q1, so the ideals aj are permuted cyclically, and on the other hand

β = (α0u)giγℓ = αgi

0 ugiγℓ, so u is changed into ug

i, giving the equivalence described in (1).

• The only fixed point of the transformation (a0, . . . , aℓ−2, u) 7→ (aℓ−2, a0, . . . , aℓ−3, ug) is obtained withall the ai equal and u = ug, but since the ai are pairwise coprime this means that they are all equal

to ZKz , and u = ugi for all i, and so u = 1.

�


Remark 2.9. Note that condition (a) implies that a ∈ (Cl(Kz)/Cl(Kz)ℓ)[T ], and for any modulus m coprime

to a also that a ∈ (Clm(Kz)/Clm(Kz)ℓ)[T ].

Lemma 2.10. Keep the above notation, and in particular recall that a =∏

0≤i≤ℓ−2 agi

i . The condition

a ∈ (I(Kz)/I(Kz)ℓ)[T ] is equivalent to the following:

(1) In the general case τ(ai) = ai−1 (equivalently, ai = τ−i(a0)), and τ (ℓ−1)/2(a0) = τ2(a0).(2) In the special case τ(ai) = ai+(ℓ−3)/2, so that a2i = τ−2i(a0) and a2i+1 = τ−2i(a1), with the following

conditions on (a0, a1):

• If ℓ ≡ 1 (mod 4) then a1 = τ (ℓ−3)/2(a0), or equivalently a0 = τ (ℓ+1)/2(a1).• If ℓ ≡ 3 (mod 4) then τ (ℓ−1)/2(a0) = a0 and τ (ℓ−1)/2(a1) = a1.

Proof. Since τ(a) =∏

i τ(ai)gi and the τ(ai) are integral, squarefree and coprime ideals, this is the canonical

decomposition of τ(a) (up to ℓth powers). On the other hand ag =∏

i agi

i−1. Assume first that we arein the general case. Since τ(a)/ag is an ℓth power, by uniqueness of the decomposition we deduce that

τ(ai) = ai−1. A similar proof using that g(ℓ−1)/2 ≡ −1 (mod ℓ) shows that τ2(ai) = ai+(ℓ−1)/2, and putting

everything together proves (1). Assume now that we are in the special case, so that τ(a)/a−g is an ℓth

power. Since −g ≡ g(ℓ+1)/2 (mod ℓ), the same reasoning shows that τ(ai) = ai−(ℓ+1)/2 = ai+(ℓ−3)/2, so in

particular τ2(ai) = ai−(ℓ+1) = ai−2, and the other formulas follow immediately. �

Definition 2.11. Let D (resp., Dℓ) be the set of all prime ideals p of k with p ∤ ℓ (resp., with p | ℓ) such thatthe prime ideals p, pz, and (in the general case) pk of K, Kz, and kz above p satisfy the following conditions:

(1) In all cases p is totally split in the extension Kz/K.(2) In the general case pk is split in the quadratic extension Kz/kz.(3) In the special case with ℓ ≡ 1 (mod 4), p is totally split in the extension Kz/k (equivalently p is split

in the quadratic extension K/k).

Note that these conditions are independent of the choices of p, pz, and pk above any particular p.

Corollary 2.12. If pz is a prime ideal of Kz dividing some ai, above a prime p of k, then p ∈ D ∩Dℓ.

Proof. Assume first that we are in the general case. Then τ acts transitively on the ai, all of which aresquarefree and coprime, and so any p dividing ai must have ℓ− 1 nontrivial conjugates (including p itself),establishing (1). Similarly, τ2(ai) = ai+(ℓ−1)/2, and for the same reason the prime ideals of Kz dividing theai come from prime ideals pk of kz which split in Kz/kz .

In the special case, if p splits as a product of h conjugate ideals in Kz, the decomposition group D(pz/p)has cardinality ef = (ℓ − 1)/h hence is the subgroup of Gal(Kz/k) generated by τh since [Kz : k] = ℓ − 1.Since τh(ai) = a(ℓ−3)h/2+i and τh fixes pz, it follows as before that (ℓ − 1) | (ℓ − 3)h/2. Now evidently(ℓ − 1, (ℓ − 3)/2) is equal to 1 if ℓ ≡ 1 (mod 4) and to 2 if ℓ ≡ 3 (mod 4). Thus when ℓ ≡ 1 (mod 4) wededuce as above that (ℓ− 1) | h hence that e = f = 1, so that p is totally split in Kz/k. On the other handif ℓ ≡ 3 (mod 4) we only have (ℓ − 1)/2 | h. If h = ℓ − 1 then p is again totally split. On the other hand,if h = (ℓ− 1)/2 then ef = 2, so p is either inert or ramified in the quadratic extension K/k, so p is totallysplit in Kz/K. �

2.3. The Mirror Field. We now introduce the mirror field of K. When ℓ = 3 this notion is classical andwell-known; the mirror field of Q(

√D) is Q(

√−3D) and the Scholz reflection principle establishes that the

3-ranks of their class groups differ by at most 1.In the case ℓ > 3 this notion is less well known but does appear in the literature (see for instance the

works of G. Gras [23] and [24]), and in particular Scholz’s theorem can be generalized to this context, seefor instance [28] for the case ℓ = 5.


Definition 2.13. In the general case, we define the mirror field K ′ of K (implicitly, with respect to the

prime ℓ) to be the degree ℓ− 1 subextension of Kz/k fixed by τ (ℓ−1)/2τ2.

We do not define the mirror field for the special case, so in this subsection we assume that we are in thegeneral case.

Lemma 2.14. Write K = k(√D) for some D ∈ k∗ \ k∗2.

(1) The extension K ′/k is cyclic of degree ℓ− 1, and K ′ = k(√

D(ζℓ − ζ−1ℓ )

).

(2) The field K ′ is a quadratic extension of k(ζℓ + ζ−1ℓ ), more precisely

K ′ = k(ζℓ + ζ−1ℓ )

(√−D(4− α2)

),

where α = ζℓ + ζ−1ℓ .

Proof. Straightforward; for (2), note that −D(4− α2) = D(ζℓ − ζ−1ℓ )2. �

The point of introducing the mirror field is the following result:

Proposition 2.15. Assume that we are in the general case. As before, let p be a prime ideal of k, pz anideal of Kz above p, and pk and p the prime ideals below pz in kz and K respectively. The following areequivalent:

(1) The ideals pk and p are both totally split in Kz/kz and Kz/K respectively (in other words p ∈ D∪Dℓ).(2) The ideal p is totally split in K ′/k.

In particular (by Corollary 2.12), (1)-(2) are true if pz divides some ai. Moreover, these conditions implythat exactly one of the following is true:

(a) p is split in K/k and totally split in kz/k.(b) p is inert in K/k and split in kz/k as a product of (ℓ− 1)/2 prime ideals of degree 2.(c) p is above ℓ, is ramified in K/k, and its absolute ramification index e(p/ℓ) is an odd multiple of

(ℓ− 1)/2 (equivalently e(p/ℓ) is an odd multiple of ℓ− 1).

Proof. (1) if and only if (2): We see that any nontrivial elements of D(pz/p) must be of the form τ iτ2 with

i 6≡ 0 (mod ℓ− 1), and squaring we have τ2i ∈ D(pz/p), so 2i ≡ 0 (mod ℓ− 1), so D(pz/p) ⊂ {1, τ (ℓ−1)/2τ2},yielding (2). The converse is proved similarly.

To prove the last statement, first recall from [12] the following result:

Lemma 2.16. Let K be any number field and Kz = K(ζℓ). The conductor of the extension Kz/K is givenby the formula

f(Kz/K) =∏

p|ℓ(ℓ−1)∤e(p/ℓ)

p .

It follows in particular that if p ∤ ℓ, or if p | ℓ and (ℓ − 1) | e(p/ℓ) then p is unramified in kz/k, andtherefore also (arguing via inertia groups) in K/k, since otherwise the ideal pk would be ramified in Kz/kz .Thus, assuming (2), the only prime ideals p which can be ramified in K/k are with p | ℓ and (ℓ− 1) ∤ e(p/ℓ).

If p is split or inert in K/k, we check that f(pz|p) equals 1 or 2 respectively, showing (a) and (b). If pis ramified, then (3.1) implies that (ℓ − 1) | e(p/ℓ) = e(p/p)e(p/ℓ). Since (ℓ − 1) ∤ e(p/ℓ) we conclude thate(p/ℓ) = n(ℓ− 1)/2 with n odd. �

The following corollaries are immediate:

Corollary 2.17. Let p be a prime ideal of k below a prime ideal pz of Kz dividing some ai as defined above.If p is ramified in the quadratic extension K/k then p is above ℓ.


Corollary 2.18. In both the general and special cases, assume that for any prime ideal p of k above ℓ theabsolute ramification index e(p/ℓ) is not divisible by (ℓ− 1)/2. Then all the ai are coprime to ℓ.

Note that for ℓ = 3 this corollary is empty, but the conclusion of the corollary always holds whenℓ > 2[k : Q] + 1, and in particular when k = Q and ℓ ≥ 5.

Proposition 2.19. There exists an ideal aα of K such that∏

0≤i≤ℓ−2 ai = aαZKz . In addition:

(1) In the general (resp., special) case, aα is stable by τ and τ2 (resp., by τ).(2) If either the assumption of Corollary 2.18 is satisfied (for instance when ℓ > 2[k : Q] + 1), or we are

in the special case with ℓ ≡ 1 (mod 4), then aα = a′αZK for some ideal a′α of k.

Proof. (1). In the general case, since τ(ai) = ai−1 we have∏

0≤i≤ℓ−2 ai = aαZKz with aα = NKz/K(a0), and

since τ2(ai) = ai+(ℓ−1)/2, aα is stable by τ2. In the special case, since τ2(ai) = ai−2 we have∏

0≤i<(ℓ−1)/2 a2i =

NKz/K(a0)ZKz and∏

0≤i<(ℓ−1)/2 a2i+1 = NKz/K(a1)ZKz , so that∏

0≤i<ℓ−1 ai = aαZKz with aα = NKz/K(a0a1)

an ideal of K, and since τ(ai) = ai+(ℓ−3)/2, aα is stable by τ .

(2). In the special case with ℓ ≡ 1 (mod 4) then (ℓ− 3)/2 is odd, so since a1 = τ (ℓ−3)/2(a0) it follows thatτ(NKz/K(a0)) = NKz/K(a1), so that

∏0≤i≤ℓ−2 ai = NKz/k(a0)ZKz = a′αZKz with a′α an ideal of the base

field k. On the other hand, if the assumption of Corollary 2.18 is satisfied then aα is coprime to ℓ, hence byCorollary 2.17 it is not divisible by any prime ramified in K/k, and since it is stable by Gal(K/k) it comesfrom an ideal a′α of k. �

3. Hecke Theory: Conductors

Our goal (see Theorem 3.8) is to give a usable expression for the “conductor” f(L) in terms of thefundamental quantities (a0, · · · , aℓ−2, u) given by Proposition 2.8, where we recall that the conductor of the

Cℓ-extension N/K is equal to f(N/K) = f(L)ZK and that d(L/k) = d(K/k)(ℓ−1)/2f(L)ℓ−1.We first recall from [12] and [10] some results concerning the cyclotomic extensions kz/k and Kz/K.

Remark 3.1. By and large we stick to the notation of [10] except that the notation m(p) of [10] is the sameas M(p) here, which corresponds to numbers Aα, while our m(p) corresponds to numbers aα.

Proposition 3.2. [10, Theorem 2.1] As above, let p be a prime of k over ℓ, and let e(p) and e(p) be therespective absolute ramification indices over ℓ. Then we have

(3.1) e(pz/p) =ℓ− 1

(ℓ− 1, e(p))and

e(pz/ℓ)

ℓ− 1=

e(p)

(ℓ− 1, e(p)).

Definition 3.3. Suppose that p, p, and pz are as above, so that e(pz/p) | (ℓ − 1). Moreover, let α ∈(K∗

z/K∗zℓ)[T ] be as in Proposition 2.4.

(1) If pZK = p2 in K/k we set p1/2 = p, and if pZKz = pe(pz/p)z in Kz/K, we set pz = p1/e(pz/p).

(2) We say that an ideal p of k divides some Gal(Kz/k)-invariant ideal b of K (resp., of Kz) when

(pZK)1/e(p/p) (resp., (pZK)1/e(pz/p)) does, or equivalently when p (resp., p1/e(pz/p)) does, where thislast condition is independent of the choice of ideal p of K above p.

(3) If e is an integer, write r(e) for the unique integer such that e ≡ r(e) (mod ℓ−1) and 1 ≤ r(e) ≤ ℓ−1.(4) We write

M(p) =ℓe(pz/ℓ)

ℓ− 1=

ℓe(p)

(ℓ− 1, e(p))∈ Z , m(p) =

M(p)

e(pz/p)=

ℓe(p)

ℓ− 1.

(5) Denote by En the congruence xℓ/α ≡ 1 (mod ∗pnz ) in Kz.(6) Define quantities Aα(p) and aα(p) as follows:

• If En is soluble for n = M(p), we set Aα(p) = M(p) + 1 and aα(p) = m(p).


• Otherwise, if n < M(p) is the largest exponent for which it is soluble, we set Aα(p) = n and wedefine

aα(p) =Aα(p)− r(e(p))/(ℓ − 1, e(p))

e(pz/p)=

⌈Aα(p)

e(pz/p)

⌉− 1 ∈ Z .

Remarks 3.4. (1) The quantity r(e(p))/(ℓ− 1, e(p)) = r(e(p))/(ℓ− 1, r(e(p))) is an integer, and equals1 when ℓ = 3 or when k = Q for instance, and the second equality for aα(p) is proved below.

(2) The notation Aα(p) and aα(p) (instead of Aα(pz) and aα(pz)) is justified by the following lemma:

Lemma 3.5. With the above assumptions, the solubility of En is independent of the ideal pz of Kz above p;i.e., it is equivalent to xℓ/α ≡ 1 (mod ∗pn/e(pz/p)) or to xℓ/α ≡ 1 (mod ∗pn/e(pz/p)).

Proof. If p′z is another ideal above p, there exists h = τ iτ j2 ∈ Gal(Kz/k) with p′z = h(pz) (resp., simply

h = τ i in the special case). Thus if xℓ/α ≡ 1 (mod ∗pnz ) we have h(x)ℓ/h(α) ≡ 1 (mod ∗p′zn). However,

since α ∈ (K∗z /K

∗zℓ)[T ], modulo ℓth powers we have τ(α) = αg and τ2(α) = α−1 (resp., τ(α) = α−g), hence

h(α) = α(−1)jgiγℓ (resp., h(α) = α(−1)igiγℓ) for some γ ∈ K∗z . We deduce that yℓ/α ≡ 1 (mod ∗p′z

n), with

y = (h(x)/γ)(−1)j g−i(resp., y = (h(x)/γ)(−1)ig−i

), proving the lemma. �

Proposition 3.6. (1) We have ℓ ∤ Aα(p), and if Aα(p) ≤ M(p) (equivalently, if Aα(p) ≤ M(p) − 1)then

Aα(p) ≡e(p)

(ℓ− 1, e(p))

(mod

ℓ− 1

(ℓ− 1, e(p))

).

(2) We have aα(p) = m(p) if Aα(p) = M(p) + 1, and otherwise

0 ≤ aα(p) ≤ℓe(p)

ℓ− 1− ℓ− 1 + r(e(p))

ℓ− 1<

ℓe(p)

ℓ− 1− 1 = m(p)− 1 .

Proof. (1) follows from Proposition 3.8 of [10], and (2) follows from the definitions and from (3.1). �

Remark 3.7. As mentioned in [14], the congruence (1), or equivalently the integrality of aα(p) (whenAα(p) < M(p)) comes from a subtle although very classical computation involving higher ramification groups;see Proposition 3.6 of [10] along with Chapter 4 of [41].

We can now quote the crucial result from [10] which gives the conductor of the extension N/K:

Theorem 3.8. [10, Theorem 3.15] Assume that (a0, . . . , aℓ−2) are as in Proposition 2.8, so that∏

0≤i≤ℓ−2 ai =

aαZKz with aα an ideal of K stable by τ2 (resp., by τ in the special case), and sometimes coming from k(see Proposition 2.19). Then the conductor of the associated field extension N/K is given as follows:

f(N/K) = ℓaα

∏p|ℓ p

⌈e(p)/(ℓ−1)⌉∏

p|ℓ , p∤aαp⌈aα(p)⌉

.

Remark 3.9. One can now draw additional conclusions about the aα(p). For example, suppose that p isa prime ideal k above ℓ with pZK = p2, p ∤ aα and aα(p) < m(p). Then vp(f(N/K)/ℓ) ≡ 0 (mod 2), asf(N/K) = f(L)ZK for an ideal f(L) of k, and it follows from the theorem and Proposition 3.6 that

(3.2) aα(p) ≡ ⌈e(p)/(ℓ − 1)⌉ (mod 2).

Definition 3.10. Let a equal either m(p), or an integer with 0 ≤ a < m(p)− 1, and define

h(0, a, p) =

{0 if (ℓ− 1) ∤ e(p) or a = m(p) ,

1 if (ℓ− 1) | e(p) and a < m(p) ;

h(1, a, p) =

{1 if (ℓ− 1) ∤ e(p) ,

2 if (ℓ− 1) | e(p) .


Remark 3.11. Note that if ℓ > 2[k : Q] + 1, for instance when k = Q and ℓ ≥ 5, we have e(p) < ℓ − 1 so(ℓ− 1) ∤ e(p). Thus in this case we simply have h(ε, a, p) = ε, independently of a and p. We will also see inRemark 4.7 that a number of other formulas simplify.

Lemma 3.12. Let p be a prime ideal of K above ℓ and denote by Cn the congruence xℓ/α ≡ 1 (mod ∗pn)in Kz. Then aα(p) is equal to the unique value of a as in the previous definition such that Cn is soluble forn = a+h(0, a, p) and not soluble for n = a+h(1, a, p), where this last condition is ignored if a+h(1, a, p) >m(p).

Proof. By Lemma 3.5 the solubility of En is equivalent to that of Cn/e(pz/p). If a = aα(p) = m(p), then En

is soluble for n = ℓe(pz/ℓ)/(ℓ− 1), which is equivalent to Cm(p) = Ca as desired.If a = aα(p) < m(p), we have Aα(p) = ae(pz/p) + r(e(p))/(ℓ − 1, e(p)), and Proposition 3.6 (1) implies

that the solubility of En for n = Aα(p) is equivalent to that of En′ when Aα(p) − (ℓ − 1)/(ℓ − 1, e(p)) <n′ ≤ Aα(p). If (ℓ − 1) ∤ e(p) we have r(e(p)) < ℓ − 1 and choose n′ = ae(pz/p), while if (ℓ − 1) | e(p) wechoose n′ = n = ae(pz/p) + 1. Thus the solubility of EAα(p) and En′ is equivalent to that of Cn′′ , wheren′′ = n′/e(pz/p) = a+ h(0, a, p) by definition of h(0, a, p). (Recall that e(pz/p) = 1 when (ℓ− 1) | e(p).)

Furthermore, since En is not soluble for n = Aα(p)+1, we also have that En′ is not soluble, where n′ = nif (ℓ− 1) | e(p) and n′ = a(pz/p) + (ℓ− 1)/(ℓ− 1, e(p)) ≥ n′ otherwise. The solubility of En′ is equivalent tothat of Cn′′ where n′′ = n′/e(pz/p) = a+ h(1, a, p), as desired.

Finally, we conclude by checking that the conditions are mutually exclusive.�

4. The Dirichlet Series

Since f(N/K) = f(L)ZK for some ideal f(L) of k, we have NK/Q(f(N/K)) = Nk/Q(f(L))2. To emphasize

the fact that we are mainly interested in the norm from k/Q, we set the following definition (norms fromextensions other than k/Q will always indicate the field extension explicitly):

Definition 4.1. If a is an ideal of k, we set N (a) = Nk/Q(a), while if a is an ideal of K, we set

N (a) = NK/Q(a)1/2 .

In particular, for each ideal a of k we have N (a) = N (aZK).

Recall that we set

Φℓ(K, s) =1

ℓ− 1+

∑

L∈Fℓ(K)

1

N (f(L))s,

with f(N/K) = f(L)ZK is given by Theorem 3.8. By Proposition 2.4, we have

(ℓ− 1)Φℓ(K, s) =∑

α∈(K∗z/K

∗zℓ)[T ]

1

N (f(L))s,

where L = Kz( ℓ√α)G (including α = 1 corresponding to L = KG

z = k with f(L) = Zk and N (f(L)) = 1), soby Proposition 2.8, we have

(ℓ− 1)Φℓ(K, s) =∑

(a0,...,aℓ−2)∈J

∑

u∈Sℓ(Kz)[T ]

1

N (f(L))s,

where J is the set of (ℓ − 1)uples of ideals satisfying condition (a) of Proposition 2.8, and f(L) is theconductor of the extension corresponding to (a0, . . . , aℓ−2, u). Thus, replacing f(L) by the formula given byTheorem 3.8, recalling that

∏p|ℓN (p)e(p) = ℓ[k:Q], and writing

e(p) = (⌈e(p)/(ℓ − 1)⌉ − 1)(ℓ− 1) + r(e(p)) ,


we obtain

(4.1) (ℓ− 1)Φℓ(K, s) = ℓ−ℓ

ℓ−1[k:Q]s

∏

p|ℓN (p)−

ℓ−1−r(e(p))ℓ−1

s∑

(a0,...,aℓ−2)∈J

Sα(s)

N (aα)s,

whereSα(s) =

∑

u∈Sℓ(Kz)[T ]

∏

p|ℓp∤aα

N (p)⌈aαu(p)⌉s ,

and where α is any element of K∗z such that α ∈ (K∗

z /K∗zℓ)[T ] and qℓ0

∏0≤i≤ℓ−2 a

gi

i = αZKz for some idealq0.

Definition 4.2. For α ∈ K∗z and an ideal b of Kz, we introduce the function

fα(b) = |{u ∈ Sℓ(Kz)[T ], xℓ/(αu) ≡ 1 (mod ∗b) soluble in Kz}| ,with the convention that fα(b) = 0 if b ∤ (1− ζℓ)

ℓZKz .

Let pi for 1 ≤ i ≤ n = n(α) be the prime ideals of k above ℓ and not dividing aα, and for each i let ai be

such that either ai = m(pi), or 0 ≤ ai ≤ m(pi)− (ℓ−1)+r(e(pi))ℓ−1 = ⌈m(pi)⌉ − 2 with ai ∈ Z, where as usual pi

is an ideal of K above pi, and let A be the set of such (a1, . . . , an). Noting that thanks to the convention of

Definition 4.1 we have∏

pi|pi N (pi) = N (pi)1/e(pi/pi), we thus have

(4.2) Sα(s) =∑

(a1,...,an)∈A

∏

1≤i≤n

N (pi)⌈ai⌉s/e(pi/pi)

∑

u∈Sℓ(Kz)[T ]∀i, aαu(pi)=ai

1 .

By Lemma 3.12, we have aαu(pi) ≥ ai if and only if u is counted by fα(pbii ), where bi = ai+h(0, a, pi), and

we rewrite pbii = pbi/e(pi/pi)i . Let B(α) be the set of n-uples (b1, . . . , bn) with 0 ≤ bi ≤ m(pi), bi ∈ Z∪{m(pi)}.

By inclusion-exclusion we obtain the following:

Lemma 4.3. We have

(4.3) Sα(s) =∑

(b1,...,bn)∈B(α)

fα

∏

1≤i≤n

pbi/e(pi/pi)i

∏

1≤i≤n

(N (pi)

⌈bi⌉s/e(pi/pi)Q(pbi/e(pi/pi)i , s)

),

where Q(pb/e(p/p), s) is defined as follows. Let as usual p be an ideal of K above p and define q = N (p)1/e(p/p).Then if b = m(p) or 0 ≤ b < m(p) with b ∈ Z:

(1) If (ℓ− 1) ∤ e(p) we set

Q(pb/e(p/p), s) =

1 if b = 0 ,

1− 1/qs if 1 ≤ b ≤ ⌈m(p)⌉ − 2 ,

−1/qs if b = ⌈m(p)⌉ − 1 ,

1 if b = m(p) .

(2) If (ℓ− 1) | e(p) we set

Q(pb/e(p/p), s) =

0 if b = 0 ,

1/qs if b = 1 ,

1/qs − 1/q2s if 2 ≤ b ≤ m(p)− 1 ,

1− 1/q2s if b = m(p) .


Remark 4.4. There are conditions on the ai, e.g. (3.2), such that the inner sum in (4.2) vanishes forimpossible choices of the ai. One can use this to prove alternate versions of Lemma 4.3 that are nonobviouslyequivalent. In particular, if (ℓ − 1) | e(p) then one can restrict to bi ∈ 2Z ∪ {m(pi)} with suitably modified

Q(pb/e(p/p), s).

Definition 4.5. (1) We let B be the set of formal products of the form

b =∏

pi|ℓ pbi/e(pi/pi)i , where the bi are such that 0 ≤ bi ≤ m(pi) and bi ∈ Z ∪ {m(pi)}.

(2) We will consider any b ∈ B as an ideal of K, where by abuse of language we accept to have fractionalpowers of prime ideals of K, and we will set bz = bZKz , which is a true ideal of Kz stable by τ , andalso by τ2 in the general case.

(3) If b ∈ B as above, we set

⌈N⌉(b) =∏

pi|bN (pi)

⌈bi⌉/e(pi/pi) and P (b, s) =∏

pi|bQ(p

bi/e(pi/pi)i , s) .

where Q(pb/e(p/p), s) := Q(pb/e(p/p), s) except in the case (ℓ − 1) | e(p) and b = 0, where we set

Q(pb/e(p/p), s) = 1.

We thus obtain

(4.4)∑

(a0,...,aℓ−2)∈J

Sα(s)

N (aα)s=

∑

b∈B⌈N⌉(b)sP (b, s)

∑

(a0,...,aℓ−2)∈J(aα,b)=1

p∤b and (ℓ−1)|e(p)⇒p|aα

fα(b)

N (aα)s.

The case p ∤ b, (ℓ− 1) | e(p), and p ∤ aα is precisely that for which Q(pb/e(p/p), s) = 0 and Q(pb/e(p/p), s) = 1.

By excluding this case we may substitute Q for Q with Q(p0, s) = 1.

Definition 4.6. (1) For b as above we define

re(b) =∏

p|ℓZK , p∤b(ℓ−1)|e(p)

p .

(2) We set dℓ =∏

p∈Dℓp (see Definition 2.11).

Remark 4.7. Since e(p) = e(p/p)e(p) ≤ 2[k : Q], we note that if ℓ > 2[k : Q]+1 then re(b) is always trivial,so that specializing to the case k = Q and ℓ ≥ 5 now would avoid some complications.

Lemma 4.8. For each aα appearing in the inner sum of (4.4) we have

(4.5) (aα, ℓZK) = re(b) =∏

p∈Dℓ(p,b)=1

∏

p|pp ,

so that re(b) | dℓ.Additionally, in the special case with ℓ ≡ 1 (mod 4) we have re(b) =

∏p∈Dℓ(p,b)=1

p.

Proof. If p ∤ b and (ℓ − 1) | e(p) then clearly p | aα. Conversely, let p | aα be above ℓ. Since (aα, b) = 1we know that p ∤ b. If we had (ℓ − 1) ∤ e(p), Proposition 3.2 would imply that e(pz/p) > 1, contradictingCorollary 2.12. This proves the first equality of (4.5), and the rest follows similarly. �


Thus we obtain

(4.6)∑

(a0,...,aℓ−2)∈J

Sα(s)

N (aα)s=

∑

b∈Bre(b)|dℓ

⌈N⌉(b)sP (b, s)∑

(a0,...,aℓ−2)∈J(aα,ℓZK)=re(b)

fα(b)

N (aα)s.

To compute fα(b) we set the following definition:

Definition 4.9. For any ideal b ∈ B, and for any subset T of Fℓ[G], we set

Sbz(Kz)[T ] = {u ∈ Sℓ(Kz)[T ], xℓ/u ≡ 1 (mod ∗bz) soluble} ,

where u is any lift of u coprime to bz, and the congruence is in Kz.

Lemma 4.10. Let (a0, . . . , aℓ−2) satisfy condition (a) of Proposition 2.8, suppose that α satisfies the condi-

tion described before Definition 4.2, and recall that we set a =∏

i agi

i . We have

fα(b) =

{|Sbz(Kz)[T ]| if a ∈ Clbz(Kz)

ℓ ,

0 otherwise.

Proof. The lemma and its proof are a direct generalization of Lemma 5.3 of [14], and we omit the details. �

5. Computation of |Sbz(Kz)[T ]|In this section we compute the size of the group Sbz(Kz)[T ] appearing in Lemma 4.10, as well as several

related quantities.

Lemma 5.1. Set Zbz = (ZKz/bz)∗. Then

|Sbz(Kz)[T ]| =|(U(Kz)/U(Kz)

ℓ)[T ]||(Clbz(Kz)/Clbz(Kz)ℓ)[T ]|

|(Zbz/Zℓbz)[T ]| ,

and in particular

|Sℓ(Kz)[T ]| = |(U(Kz)/U(Kz)ℓ)[T ]||(Cl(Kz)/Cl(Kz)

ℓ)[T ]| .Proof. This is a minor variant of Corollary 2.13 of [10], proved in the same way. �

The quantity |(U(Kz)/U(Kz)ℓ)[T ]| is given by the following lemma.

Lemma 5.2. Assume that ℓ > 3, the case ℓ = 3 being treated in [14, Lemma 5.4]. For any number field M ,write rkℓ(U(M)) := dimFℓ

(U(M)/U(M)ℓ), and denote by r1(M) and r2(M) the number of real and pairs ofcomplex embeddings of M .

(1) For any number field M we have

rkℓ(U(M)) =

{r1(M) + r2(M)− 1 if ζℓ /∈ M ,

r1(M) + r2(M) if ζℓ ∈ M .

(2) We have |(U(Kz)/U(Kz)ℓ)[T ]| = ℓRU(K), where

RU(K) :=

r2(K)− r2(k) in the general case,

r1(k) + r2(k) in the special case with ℓ ≡ 3 (mod 4) ,

r2(k) in the special case with ℓ ≡ 1 (mod 4) .

(3) In particular, if k = Q we have RU(K) = r2(K) in all cases.


Proof. (1) is Dirichlet’s theorem, and (3) is a consequence of (2). To prove (2) in the general case, whereT = {τ2 + 1, τ − g}, we apply the exact sequence

(5.1) 1 −→ U(kz)

U(kz)ℓ[τ − g] −→ U(Kz)

U(Kz)ℓ[τ − g] −→ U(Kz)

U(Kz)ℓ[τ2 + 1, τ − g] −→ 1 ,

where the last nontrivial map sends ε to τ2(ε)/ε. Surjectivity follows from Lemma 2.3, and (τ2+1)(τ2−1) = 0implies that the two nontrivial maps compose to zero. Finally, suppose ε ∈ U(Kz) satisfies τ2(ε) = εηℓ forsome η ∈ Kz. Applying τ2 to both sides we see that ητ2(η) = ζaℓ for some a, and replacing η with η1 = ηζbℓwith a + 2b ≡ 0 (mod ℓ), we obtain η1τ2(η1) = 1 and τ2(ε) = εηℓ1. By Hilbert 90 there exists η2 withη1 = η2/τ2(η2), so that ε1 = εηℓ2 satisfies τ2(ε1) = ε1, in other words ε1 ∈ kz, proving exactness of (5.1).

By a nontrivial theorem of Herbrand (see Theorem 2.3 of [10]), we have |(U(Kz)/U(Kz)ℓ)[τ−g]| = ℓr2(K)+1

and |(U(kz)/U(kz)ℓ)[τ − g]| = ℓr2(k)+1, establishing (2) in the general case.

In the special case, with T = {τ + g} = {τ − g(ℓ+1)/2}, (2) follows directly from Herbrand’s theoremapplied to the extension kz/k = Kz/k, for which τ generates the Galois group. �

Note that for ℓ = 3 the same is true except that in the special case we have RU(K) = r1(k) + r2(k)− 1.This follows from the shape of [10, Theorem 2.3], or may be easily verified from [14, Lemma 5.4].

Lemma 5.3. Let b ∈ B satisfy bz | (1− ζℓ)ℓ, and define cz =

∏pz⊂Kz

pz |bzp⌈vpz (bz)/ℓ⌉z . We have

|(Zbz/Zℓbz)[T ]| = |(cz/bz)[T ]| ,

the latter being considered as an additive group.

Proof. See Proposition 2.6 and Theorem 2.7 of [10], or Lemma 1.5.6 of [34]. �

Theorem 5.4. We have in the general case

(5.2) |(cz/bz)[τ − gj ]| =∏

p|bzNK/Q(p)

xj (p) ,

where

(5.3) xj(p) =⌈vp(b)−

je(p)

ℓ− 1

⌉−

⌈⌈e(pz/p)vp(b)/ℓ⌉(ℓ − 1, e(p)) − je(p)

ℓ− 1

⌉.

In the special case, (5.3) holds with p and K replaced throughout by p and k respectively.Finally, in the general case, then (5.3) is also true with respect to kz/k. In this case one must replace p,

K, bz, and cz respectively by p, k, bk := cz ∩ kz, and

(5.4) ck := cz ∩ Zkz =∏

pk⊂kzpk|bk

p⌈vpk (bk)/ℓ⌉k .

Proof. This is the result at the bottom of [10, p. 177], applied to Kz/K, Kz/k, and kz/k respectively. As in

[10, Theorem 2.7] the result may be simplified if vp(b) is either an integer or equal to ℓe(p)ℓ−1 , and in particular

always in the general case with respect to Kz/K, but in other cases vp may be a half integer.Finally, the equality in (5.4) is readily verified. �

Recall from [10, Theorem 2.1] and (3.1) that e(pz/p) =ℓ−1

(ℓ−1,e(p)) and e(pk/p) =ℓ−1

(ℓ−1,e(p)) . In the special

case, this theorem together with Lemma 5.3 gives the cardinality of (Zbz/Zℓbz)[T ] by choosing j = (ℓ+1)/2.

In the general case we require the following additional lemma:


Lemma 5.5. Assume that we are in the general case and set ck = cz ∩ kz and bk = bz ∩ kz. We have

|(Zbz/Zℓbz)[T ]| = |(cz/bz)[τ − g]|/|(ck/bk)[τ − g]| ,

where the two terms on the right-hand side are given by Theorem 5.4.

Proof. We have an exact sequence of Fℓ[G]-modules

1 −→ cz

bz[τ2 − 1][τ − g] −→ cz

bz[τ − g] −→ cz

bz[T ] −→ 1 ,

the last map sending x to x− τ2(x). It therefore suffices to argue that (cz/bz)[τ2−1] = (cz ∩kz)/(bz ∩kz): ifx ∈ cz satisfies τ2(x) = x+ y for some y ∈ bz, then applying τ2 we see that τ2(y) = −y, hence τ2(x+ y/2) =x+ y/2. Moreover x+ y/2 ≡ x (mod bz), because 2 is invertible modulo ℓ hence modulo b. �

Definition 5.6. We set Gb = (Clbz(Kz)/Clbz (Kz)ℓ)[T ].

We conclude with one additional lemma which will be needed in the next section.

Lemma 5.7. In the general case set u = ι(τ2 + 1)ι(τ − g) and in the special case set u = ι(τ + g).

(1) The map I 7→ u(I) induces a surjective map from Clbz(Kz)/Clbz(Kz)ℓ to Gb, of which a section is

the natural inclusion from Gb to Clbz(Kz)/Clbz(Kz)ℓ.

(2) Any character χ ∈ Gb can be naturally extended to a character of Clbz(Kz)/Clbz(Kz)ℓ by setting

χ(I) = χ(u(I)), which we again denote by χ by abuse of notation.

(3) Let as usual a =∏

0≤i≤ℓ−2 agi

i with the ai satifying condition (a) of Proposition 2.8.

• In the general case and in the special case when ℓ ≡ 1 (mod 4), we have χ(a) = χ(a0)−1;

• In the special case when ℓ ≡ 3 (mod 4), we have χ(a) = χ(a0ag1)

(ℓ−1)/2,where χ on the right-hand side is defined in (2).

Proof. (1) and (2) are immediate from Lemma 2.3. For (3), assume that we are in the special case. Using

Lemma 2.10 we have a2i = τ−2i(a0), a2i+1 = τ−2i(a1), and χ(τ2(I)) = χ(I)g2, so that

χ(a) =∏

0≤i<(ℓ−1)/2

χ(τ−2i(a0ag1))

g2i =∏

0≤i<(ℓ−1)/2

χ(a0ag1) = χ(a0a

g1)

(ℓ−1)/2 .

If in addition ℓ ≡ 1 (mod 4) we have a1 = τ (ℓ−3)/2(a0) and χ(τ(I)) = χ(I−g), giving χ(a1) = χ(a0)(−g)(ℓ−3)/2

=

χ(a0)−g(ℓ−3)/2

and χ(ag1) = χ(a0), so χ(a0ag1)

(ℓ−1)/2 = χ(a0)ℓ−1 = χ(a0)

−1.The general case of (3) is proved similarly, with ai = τ−i(a0).

�

6. Semi-Final Form of the Dirichlet Series

We can now put everything together, and obtain a complete analogue of the main theorem of [14]:

Theorem 6.1. We have

Φℓ(K, s) =ℓRU(K)

(ℓ− 1)ℓℓ

ℓ−1[k:Q]s

∏

p|ℓN (p)−

(ℓ−1−r(e(p))ℓ−1

s ·

·∑

b∈Bre(b)|dℓ

( ⌈N⌉(b)N (re(b))

)s P (b, s)

|(Zbz/Zℓbz)[T ]|

∑

χ∈Gb

F (b, χ, s) ,


where

F (b, χ, s) =∏

p|re(b)p∈D′

ℓ(χ)

(ℓ− 1)∏

p|re(b)p∈Dℓ\Dℓ

′(χ)

(−1)∏

p∈D′(χ)

(1 +

ℓ− 1

N (p)s

) ∏

p∈D\D′(χ)

(1− 1

N (p)s

),

and D′(χ) (resp. D′ℓ(χ)) is the set of p ∈ D (resp. Dℓ) such that χ(pz) = 1, where pz is any prime ideal of

Kz above p.

Proof. We begin with the formula for Φℓ(K, s) given by (4.1) and (4.6). By Remark 2.9 we have a ∈(Clbz(Kz)/Clbz(Kz)

ℓ)[T ] with a =∏

0≤i≤ℓ−2 agi

i . Thus a ∈ Clbz(Kz)ℓ if and only if χ(a) = 1 for all characters

χ ∈ Gb. The number of such characters being equal to |Gb|, by orthogonality of characters and Lemmas4.10, 5.1, and 5.2 we obtain


(ℓ− 1)ℓℓ

ℓ−1[k:Q]s

∏

p|ℓN (p)−

ℓ−1−r(e(p))ℓ−1

s∑

b∈Bre(b)|dℓ

⌈N⌉(b)sP (b, s)

|(Zbz/Zℓbz)[T ]|

∑

χ∈Gb

H(b, χ, s) ,

with

H(b, χ, s) =∑

(a0,··· ,aℓ−2)∈J ′

(aα,ℓZK)=re(b)

χ(a)

N (aα)s,

where aα was defined in Proposition 2.19, and J ′ is the set of (ℓ−1)uples of coprime squarefree ideals of Kz,satisfying condition (a) of Proposition 2.8, but now without the condition that the ideal class of a belongsto Cl(Kz)

ℓ, so satisfying the condition of Lemma 2.10.Assume first that we are in the general case. By Lemma 2.10 we can replace the sum over J ′ by a

sum over ideals a0 of Kz. The conditions and quantities linked to a0 are then as follows:

(a) The ideal a0 is a squarefree ideal of Kz such that τ (ℓ−1)/2(a0) = τ2(a0).(b) The ideals a0 and τ i(a0) are coprime for (ℓ− 1) ∤ i.(c) If pz is a prime ideal of Kz dividing a0, p the prime ideal of K below pz, and p the prime ideal of k below

pz then by Corollary 2.12 we have p ∈ D∪Dℓ. Conversely, if this is satisfied then the ideals ai = τ−i(a0)must be pairwise coprime since otherwise aα would be divisible by some p2z which is impossible since p

is unramified in Kz/K.(d) We have NKz/K(a0) = aα.

(e) By Lemma 5.7 we have χ(a) = χ−1(a0).

Thus if we denote temporarily by J ′′ the set of ideals a0 of Kz satisfying the first three conditions above,we have

H(b, χ, s) =∑

a0∈J ′′

(NKz/K(a0),ℓZK)=re(b)

χ−1(a0)

N (NKz/K(a0))s.

So that we can use multiplicativity, write a0 = cd, where c is the ℓ-part of a0 and d is the prime to ℓ part(recall that a0 is squarefree). The condition (NKz/K(a0), ℓZK) = re(b) is thus equivalent toNKz/K(c) = re(b).Thus H(b, χ, s) = ScSd with

Sc =∑

c∈J ′′

NKz/K(c)=re(b)

χ−1(c)

N (NKz/K(c))sand Sd =

∑

d∈J ′′

(NKz/K(d),ℓZK)=1

χ−1(d)

N (NKz/K(d))s.


Consider first the sum Sd. By multiplicativity we have Sd =∏

p∈D Sd,p with

Sd,p =∑

d|pZKz

τ (ℓ−1)/2(d)=τ2(d)

χ−1(d)

N (NKz/K(d))s.

As p is not above ℓ, it is unramified in K/k by Proposition 2.15 and we consider the remaining two cases:

(1) Assume that pZK = p, i.e, that p is inert in K/k. Since p is totally split in Kz/K we havepZKz =

∏0≤i≤ℓ−2 τ

i(pz) for some prime ideal pz of Kz. Furthermore, since pz/pk (with our usual

notation) is split we have τ2(pz) 6= pz, and since p is stable by τ2, τ2(pz) is again above p, so we

deduce that τ2(pz) = τ (ℓ−1)/2(pz).Since d is squarefree and coprime to its Kz/K-conjugates, we see that d = ZKz or d = τ i(pz) for

some i, with N (NKz/K(d)) equal to 1 or N (p) respectively. In the latter case we have

(6.1) Sd,p = 1 +∑

0≤i≤ℓ−2

χ(pz)−gi

N (p)s= 1 +

∑

1≤j≤ℓ−1

χ(pz)j

N (p)s,

so that Sd,p = 1 + (ℓ− 1)/N (p)s if χ(pz) = 1, and Sd,p = 1− 1/N (p)s otherwise.

(2) If instead pZK = pτ2(p) is split in K/k, then similarly either d = ZKz or d = τ i(pzτℓ−12 τ2(pz)) for

some i and pz . We have that χ(τ (ℓ−1)/2(τ2(pz))) = χ−1(τ2(pz)) = χ(pz), and hence obtain the sameresult as above.

Consider now the sum Sc. By multiplicativity, since b is stable by τ2, and applying Lemma 4.8 we have

Sc =1

N (re(b))s

∑

c∈J ′′

NKz/K(c)=re(b)

χ−1(c) =1

N (re(b))s

∏

p∈Dℓ(p,b)=1

Sc,p ,

with

Sc,p =∑

c|pZKz

τ (ℓ−1)/2(c)=τ2(c)NKz/K(c)=

∏p|p p

χ−1(c) .

Our analysis is essentially the same as before, except p can now be ramified in K/k and the possibilityc = ZKz is now excluded. In all cases we obtain that Sc,p = ℓ− 1 if χ(pz) = 1 and −1 otherwise.

Putting everything together proves the theorem in the general case.

In the special case with ℓ ≡ 1 (mod 4) the proof is similar; condition (a) is absent and (d) becomesNKz/k(a0) = N (aα). Imitating the inert case of the previous argument, we obtain the same results.

In the special case with ℓ ≡ 3 (mod 4), we replace the sum over J ′ by a sum over pairs (a0, a1) ofideals of Kz satisfying suitable conditions:

• In place of (a), a0 and a1 are fixed by τ (ℓ−1)/2.• In place of (b), the ideals a0, a1, τ

2i(a0), and τ2i(a1) must all be coprime.• In place of (d), we have NKz/K(a0a1) = N (aα).

• In place of (e), we have χ(a) = χ(a0ag1)

(ℓ−1)/2.


We must again consider all splitting types in K/k, and the arguments are similar. If p is inert, we computethat

Sd,p = 1 +∑

0≤i≤ ℓ−32

χ(pz)g2i·(ℓ−1)/2

N (p)s+

∑

0≤i≤ ℓ−32

χ(pz)g2i+1·(ℓ−1)/2

N (p)s,

equal to the same expression as before. If p is split, recall that by Proposition 2.19 aα must be stable by τ ;the relevant computation is

χ(pzτ (ℓ−1)/2(pz))(ℓ−1)/2 = χ(p1−g(ℓ−1)/2

z )(ℓ−1)/2 = χ(pz)−1,

and again we obtain the same results. For p ∈ Dℓ the argument is similar, once again considering all threecases and obtaining the same result. �

As mentioned in Remarks 3.11, if ℓ > 2[k : Q] + 1, and in particular if k = Q and ℓ ≥ 5, we always havere(b) = (1). The theorem simplifies and gives the following:

Corollary 6.2. Keep the same notation, and assume that ℓ > 2[k : Q] + 1. We have


(ℓ− 1)ℓℓ

ℓ−1[k:Q]s

∏

p|ℓN (p)−

ℓ−1−r(e(p))ℓ−1

s∑

b∈B

⌈N⌉(b)sP (b, s)

|(Zbz/Zℓbz)[T ]|

∑

χ∈Gb

F (b, χ, s) ,

where

F (b, χ, s) =∏

p∈D′(χ)

(1 +

ℓ− 1

N (p)s

) ∏

p∈D\D′(χ)

(1− 1

N (p)s

).

In the general case, we now prove that the group Gb can be described in somewhat simpler terms, interms of the mirror field K ′ of K. (See also Theorems 9.1 and Theorem 9.7 for a further characterization.)

Proposition 6.3. There is a natural isomorphism

Clb(Kz)

Clb(Kz)ℓ[T ] → Clb′(K

′)Clb′(K ′)ℓ

[τ − g],

where b′ = b ∩K ′.Moreover, using this isomorphism to regard a character χ of Clb(Kz)

Clb(Kz)ℓ[T ] as a character χ of

Clb′(K′)

Clb′(K′)ℓ[τ−g],

the condition χ(pz) = 1 defining D(χ)∩D′ℓ(χ) is equivalent to the condition χ(pK ′) = 1 for the unique prime

pK ′ of K ′ below pz.

Proof. The first statement is also proved in [15, Proposition 3.6], so we will be brief. As τ (ℓ−1)/2τ2 actstrivially on Gb, it can be checked that elements of Gb can be represented by an ideal of the form aτ2τ

(ℓ−1)/2a,which is of the form a′ZKz for some ideal a′ of K ′. We therefore obtain a well-defined injective mapClb(Kz)Clb(Kz)ℓ

[T ] → Clb′(K′)

Clb′(K′)ℓ[τ − g], which may easily be shown to be surjective as well.

The latter statement follows because the condition χ(pK ′) = 1 is equivalent to χ(pK ′ZKz) = 1, which iseasily seen to be equivalent to χ(pz) = 1 for any splitting type of pz|pK ′ .

�

7. Specialization to k = Q

We now specialize to k = Q, where we will obtain more explicit results. Henceforth we assume thatK = Q(

√D) is a quadratic field with discriminant D, with r2(D) = 0 if D > 0 and r2(D) = 1 if D < 0.


By definition, B = {1, (ℓ), (ℓ)ℓ/(ℓ−1)} in the general case with ℓ ∤ D, and B = {1, (ℓ)1/2, (ℓ), (ℓ)ℓ/(ℓ−1)} inthe special case or in the general case with ℓ | D. Equivalently we may write

(7.1) bz ∈{ZKz , (1− ζℓ)

(ℓ−1)/2ZKz , (1− ζℓ)ℓ−1ZKz = ℓZKz , (1− ζℓ)

ℓZKz

},

with the second entry removed in the former case. Throughout, (−,−,−,−) will describe quantities de-pending on B, with asterisks denoting ‘not applicable’.

Proposition 7.1. We have that |(Zbz/Zℓbz)[T ]| is equal to (1, ∗, ℓ, ℓ) or (1, 1, ℓ, ℓ) for ℓ ∤ D or ℓ|D respectively,

unless ℓ = 3 in the special case, in which case |(Zbz/Z3bz)[T ]| = (1, 1, 1, 3).

Proof. This follows from Theorem 5.4 and Lemma 5.5. In the general case we obtain

|(Zbz/Zℓbz)[T ]| = (0, ∗, 2, 2) − (0, ∗, 1, 1) = (0, ∗, 1, 1) ,

|(Zbz/Zℓbz )[T ]| = (0, 1, 2, 2) − (0, 1, 1, 1) = (0, 0, 1, 1) ,

depending on whether ℓ ∤ D or ℓ|D respectively; in the special case we obtain

|(Zbz/Zℓbz)[T ]| = (0, 0, 1, 1) ,

|(Zbz/Zℓbz)[T ]| = (0, 0, 0, 1) ,

depending on whether ℓ ≥ 5 or ℓ = 3 respectively. �

Recall by Lemma 2.14 that the mirror field of K = Q(√D) with respect to ℓ is the degree ℓ − 1 field

K ′ = Q(√D(ζℓ − ζ−1

ℓ )). The following is immediate from the results of Section 2:

Lemma 7.2. Let p be a prime different from ℓ.

• We have p ∈ D if and only if p ≡(Dp

)(mod ℓ).

• In the general case, this is equivalent to p splitting completely in K ′/Q.• In the special case with ℓ ≡ 1 (mod 4), this is equivalent to p ≡ 1 (mod ℓ).• In the special case with ℓ ≡ 3 (mod 4), this is equivalent to p ≡ ±1 (mod ℓ).

We come now to the analogue of Theorem 3.2 of [16]. The case ℓ = 3, which is slightly different, is treatedin loc. cit.:

Theorem 7.3. Assume that ℓ ≥ 5 and let K = Q(√D). We have

Φℓ(K, s) =ℓr2(D)

ℓ− 1

∑

b∈BAb(s)

∑

χ∈Gb

F (b, χ, s) ,

where the Ab(s) are given by the following table:

Condition on D A(1)(s) A(√

(−1)(ℓ−1)/2ℓ)(s) A(ℓ)(s) A(ℓℓ/(ℓ−1))(s)

ℓ ∤ D ℓ−2s 0 −ℓ−2s−1 1/ℓ

ℓ | D ℓ−3s/2 ℓ−s − ℓ−3s/2 −ℓ−s−1 1/ℓ

F (b, χ, s) =∏

p≡(Dp) (mod ℓ), p 6=ℓ

(1 +

ωχ(p)

ps

),

where we set:

ωχ(p) =

{ℓ− 1 if χ(pz) = 1

−1 if χ(pz) 6= 1 ,

where as usual pz is any ideal of Kz above p.


Proof. The computation is routine, given the following consequences of our previous results:

• We have k = Q so ℓℓ

ℓ−1[k:Q]s = ℓℓs/(ℓ−1).

• The factor∏

p|ℓN (p)... is equal to ℓ−(ℓ−2)s/(ℓ−1)) if ℓ ∤ D and to ℓ−(ℓ−3)s/(2(ℓ−1)) if ℓ | D. Multiplied

by the first factor this gives ℓ−2s if ℓ ∤ D and ℓ−3s/2 if ℓ | D.

• We have ℓRU(K) = ℓr2(D) by Lemma 5.2, with r2(D) := r2(Q(√D)).

• By Definitions 4.5 and 4.1, we have ⌈N⌉(b) = (1, ∗, ℓ, ℓ2) and ⌈N⌉(b) = (1, ℓ1/2, ℓ, ℓ3/2) for ℓ ∤ D andℓ | D respectively.

• As already mentioned, if k = Q and ℓ > 3 we have re(b) = (1), so the terms and conditions involvingre(b) disappear (in other words we use Corollary 6.2).

• By Lemma 7.2, we have p ∈ D if and only if p ≡(Dp

)(mod ℓ) and p 6= ℓ, and Dℓ = ∅ when ℓ 6= 3 by

what we have just said.• By Lemma 4.3 and Definition 4.5 of P (b, s), when ℓ ∤ D and ℓ | D respectively. we have P (b, s) =

(1, ∗,−ℓ−s, 1), and P (b, s) = (1, 1 − ℓ−s/2,−ℓ−s/2, 1) for ℓ ∤ D respectively for the usual sequence ofb.

• The values of |(Zbz/Zℓbz)[T ]| are given in Proposition 7.1.

�

Corollary 7.4. Assume that ℓ ≥ 5, and set Lℓ(s) = 1 + (ℓ − 1)/ℓ2s if ℓ ∤ D and Lℓ(s) = 1 + (ℓ − 1)/ℓs ifℓ | D. There exists a function φD(s) = φD,ℓ(s), holomorphic for ℜ(s) > 1/2, such that

Φℓ(K, s) = φD(s) +1

(ℓ− 1)ℓ1−r2(D)Lℓ(s)

∏


(1 +

ℓ− 1

ps

).

Proof. Same as in Proposition 7.5 of [14]: The main term is the contribution of the trivial characters, and

φD(s) is the contribution of the nontrivial characters: we first regard each χ ∈ Gb as a character ofClb′(K

′)Clb′(K

′)ℓ

by Proposition 6.3 and then by setting χ equal to 1 on the orthogonal complement ofClb′(K

′)

Clb′(K′)ℓ[τ − g]. By

the previous lemma, the primes occurring in the product are precisely those for which p is totally split in

K ′. Therefore, for each set of nontrivial characters χ, χ2, . . . , χℓ−1 ∈ Gb, the sum of products F (b, χ, s) maybe written as g(s) +

∑χ L(s, χ), where L(s, χ) is the (holomorphic) Hecke L-function associated to χ, and

g(s) is a Dirichlet series supported on squarefull numbers, absolutely convergent and therefore holomorphicin ℜ(s) > 1/2. Therefore φD(s) is holomorphic in ℜ(s) > 1/2 as well. We also note that the product ofthe main term may similarly be written as h(s) +L(s, ω0), where ω0 is the trivial Hecke character, and h(s)satisfies the same properties as g(s).

The ℓ = 3 case is slightly different due to the nontriviality of re(b); see [14]. �

Corollary 7.5. Assume that ℓ ≥ 5 and denote by Mℓ(D;X) the number of L ∈ Fℓ(Q(√D)) such that

f(L) ≤ X. Set c1(ℓ) = 1/((ℓ− 1)ℓ1−r2(D)), c2(ℓ) = (ℓ2 + ℓ− 1)/ℓ2 when ℓ ∤ D or c2(ℓ) = 2− 1/ℓ when ℓ | D.

(1) In the general case, or in the special case with ℓ ≡ 1 (mod 4), for any ε > 0 we have

Mℓ(D;X) = Cℓ(D)X +OD(X1− 2

ℓ+3+ε) , with

Cℓ(D) = c1(ℓ)c2(ℓ)Ress=1

∏

p≡(Dp) (mod ℓ)

(1 +

ℓ− 1

ps

),

and in the special case the product is equivalently over p ≡ 1 (mod ℓ).(2) In the special case with ℓ ≡ 3 (mod 4), for any ε > 0 we have

Mℓ(D;X) = Cℓ(D)(X log(X) + C ′ℓ(D)) +OD(X

1− 2ℓ+3

+ε) , with


Cℓ(D) = c1(ℓ)c2(ℓ) lims→1+

(s− 1)2∏

p≡±1 (mod ℓ)

(1 +

ℓ− 1

ps

),

and C ′ℓ(D) can also be given explicitly if desired.

Proof. The result follows by the same proof as in [14], with Cℓ(D) equal to the residue at s = 1 of Φℓ(K, s).We briefly recall how to obtain the error term. By the proof of Corollary 7.4, it equals (up to an implied

constant depending on D and ℓ) the error made in estimating partial sums of Hecke L-functions of degreeℓ − 1. We carry this out in the standard way, subject to the limitation that we may not shift any contourto ℜ(s) ≤ 1/2. We have by Perron’s formula, for each Hecke L-function ξ(s) =

∑n a(n)n

−s and any c > 1,

∑

n<X

a(n) =1

2πi

∫ c+i∞

c−i∞ξ(s)

Xs

sds ,

and we shift the portion of the contour from c− iT to c+ iT to ℜ(s) = σ for σ ∈ (1/2, 1) and T > 0 to be

determined. By convexity we have |ξ(s)| ≪ Tℓ−12

(1−σ+ε), and choosing c = 1 + ε, σ = 1/2 + ε our integral

is ≪ Tℓ−12

( 12+2ε)X

12+ε + X1+2ε

T ; then choosing T = X2

ℓ+3 we obtain an error term of X1− 2ℓ+3

+ε. �

In a separate paper by the first author [9], one explains how to compute the constants Cℓ(D) to highaccuracy (100 decimal digits, say) for reasonably small values of |D|. For example, we have

C3(−3) = 0.0669077333013783712918416 · · · , C3(−4) = 0.1362190676241212841449867 · · · .

8. Study of the Groups Gb

In this section, where we continue to assume that k = Q and also assume that ℓ ≥ 5, we study the groupsGb appearing in Theorem 7.3. Much of this was carried out in our paper [15] with Rubinstein-Salzedo, andwe give only a brief account of those results which are proved there.

We are indebted to Hendrik Lenstra for help in this section.

We recall a few of the important notations used previously:

• Kz is an abelian extension of Q containing the ℓth roots of unity, with G = Gal(Kz/Q) = 〈τ, τ2〉 or〈τ〉 in the general and special cases respectively.

• As in Proposition 2.4, Nz = Kz( ℓ√α) is a cyclic extension, for which we wrote αZKz = qℓ

∏0≤i≤ℓ−2 a

gi

i

and (in Proposition 2.19)∏

0≤i≤ℓ−2 ai = aαZKz for an ideal aα of K.

• We recall the possibilities for b (equivalently, bz) from (7.1), and we continue to use the notation(−,−,−,−) for quantities depending on b.

For any b as in (7.1) we define b∗ := (1− ζℓ)ℓ/bz.

Proposition 8.1. With the notation above, we have f(Nz/Kz) | bz if and only if α ∈ Sb∗(Kz).

Proof. This is very classical, and essentially due to Kummer and Hecke: for instance, by Theorem 3.7 of [10]we have

f(Nz/Kz) = (1− ζℓ)ℓaα/

∏

pz |ℓ, pz ∤aα

pAα(pz)−1z .

Thus, since aα is coprime to the product then f(Nz/Kz) | (1 − ζℓ)ℓ if and only if aα = ZK , i.e., if and only

if α is a virtual unit. If this is the case, then f(Nz/Kz) | bz if and only if the product is a multiple of(1 − ζℓ)

ℓ/bz = b∗, and by the definition of Aα and the congruence in Proposition 3.6, this is equivalent tothe solubility of the congruence xℓ/α ≡ 1 (mod ∗b∗), hence to α ∈ Sb∗(Kz). �


Theorem 8.2. [15, Corollary 3.2] Writing Cb := Clbz(Kz)/Clbz(Kz)ℓ, so that Gb = Cb[T ], and µℓ for the

group of ℓth roots of unity, there exists a perfect, G-equivariant pairing of Fℓ[G]-modules

Cb × Sb∗(Kz) 7→ µℓ .

Proof. This is the Kummer pairing: given a ∈ Cb, let σa denote its image under the Artin map; givenα ∈ Sb∗(Kz), let α be any lift; then define the pairing by (a, α) 7→ σa( ℓ

√α)/ ℓ

√α ∈ µℓ. �

Corollary 8.3. [15, Corollary 3.3 (in part)] In the general case, where T = {τ − g, τ2 + 1}, define T ∗ ={τ − 1, τ2 + 1}, and in the special case, where T = {τ + g}, define T ∗ = {τ + 1}. Then we have a perfectpairing

Gb × Sb∗(Kz)[T∗] 7→ µℓ .

In particular, we have|Gb| = |Sb∗(Kz)[T

∗]| .Proof. Recalling that τ(ζℓ) = ζgℓ , for any j the preceding corollary yields a perfect pairing

Cb[τ − gj ]× Sb∗(Kz)[τ − g1−j ] 7→ µℓ .

We conclude by taking j = 1 and j = (ℓ+ 1)/2 in the general and special cases respectively. �

Proposition 8.4. In the special case, we have Cl(Qz)/Cl(Qz)ℓ[τ + 1] = {1} .

Proof. We first show that there exists an isomorphism

Cl(Qz)/Cl(Qz)ℓ[τ + 1] ≃ Cl(K)/Cl(K)ℓ[τ + 1] .

By Lemma 2.3 (which also applies to t = τ + 1), the left side consists of those classes which may berepresented by ideals of the form NQz/K(a)/τ(NQz/K(a)). We therefore obtain a well-defined, injective map

to Cl(K)/Cl(K)ℓ[τ + 1]. Any ideal in the target space may be represented by an ideal of the form c/τ(c),

which is equivalent to (c/τ(c))(ℓ−1)2 , and c(ℓ−1)2 = NQz/K(c2(ℓ−1)ZQz), so that the map is surjective as well.

Now it suffices to show that ℓ ∤ h(±ℓ), where h(D) denotes the class number of Q(√D), and this follows

from the fact that h(±ℓ) < ℓ for all prime ℓ. �

Remark 8.5. For ℓ ≡ 3 (mod 4) it is also possible to prove the proposition via the Herbrand-Ribet theoremand a congruence for Bernoulli numbers.

Now suppose that ℓ ≡ 1 (mod 4). Then the Ankeny-Artin-Chowla conjecture (AAC) [1, 35] states that if

ǫ = (a + b√ℓ)/2 is the fundamental unit of Q(

√ℓ), then ℓ ∤ b. We will use the statement of the conjecture

directly, but we note that Ankeny and Chowla [2] and Kiselev [27] proved that it is equivalent to the conditionℓ ∤ B(ℓ−1)/2, which is trivially true if ℓ is a regular prime, a result first proved by Mordell [35]. It has been

verified for ℓ ≤ 2 ·1011 by van der Poorten, te Riele, and Williams [40], but as mentioned in the introduction,on heuristic grounds it it probably false.

Lemma 8.6. (1) If AAC is true for ℓ, the congruence xℓ ≡ ε (mod (1 − ζ)kZQz) is solvable for k =(ℓ− 1)/2, and not for any larger value of k.

(2) If AAC is false for ℓ, then this congruence is soluble for all k.

Proof. First assume AAC, and write ε = (a+b√ℓ)/2 with a, b ∈ Z. Note first that (1−ζ)(ℓ−1)/2ZQz =

√ℓZQz ,

and ε ≡ a/2 ≡ (a/2)ℓ (mod√ℓZQz), so the congruence is solvable with k = (ℓ− 1)/2.

Conversely for each x ∈ ZQz we have xℓ ≡ m (mod ℓ) for some m ∈ Z, so that a solution to xℓ ≡ ε

(mod√ℓ(1− ζ)ZQz) would yield a+ b

√ℓ ≡ 2m (mod

√ℓ(1− ζ)ZQz). We thus have a ≡ 2m (mod

√ℓ) and

hence a ≡ 2m (mod ℓ), yielding b ≡ 0 (mod (1− ζ)ZQz) and so ℓ | b, violating AAC and proving (1).To obtain (2), observe that the congruence may trivially be solved for k = 3(ℓ − 1)/2 with x ∈ Z, after

which a Newton-Hensel iteration as in [8, Lemma 10.2.10] settles the matter. �


We now return to the groups Sb∗(Kz)[T∗].

Proposition 8.7. (1) In the general case we have Sb∗(Kz)[T∗] ≃ Sb∗∩K(K).

(2) In the special case with ℓ ≡ 3 (mod 4), we have Sb∗(Kz)[T∗] = {1} for all b.

(3) In the special case with ℓ ≡ 1 (mod 4), if the Ankeny-Artin-Chowla conjecture is true for ℓ, thenwe have |Sb∗(Kz)[T

∗]| = (1, 1, ℓ, ℓ) for b as in (7.1). If Ankeny-Artin-Chowla is false for ℓ, then wehave instead |Sb∗(Kz)[T

∗]| = (ℓ, ℓ, ℓ, ℓ).

Proof. (1) [15, Proposition 3.4]. We have an injection Sb∗∩K(K) −→ Sb∗(Kz)[τ − 1] which we prove issurjective by Hilbert 90 and some elementary computations, yielding an isomorphism Sb∗(Kz)[τ−1, τ2+1] ≃Sb∗∩K(K)[τ2 + 1]. Furthermore, we have

Sb∗∩K(K) = Sb∗∩K(K)[τ2 + 1]⊕ Sb∗∩K(K)[τ2 − 1] ,

and we argue that Sℓ(K)[τ2 − 1] is trivial (and a fortiori all the Sb∗∩K [τ − 1]), again using Hilbert 90.

(2) and (3). Assume now that we are in the special case, so that Kz = Qz = Q(ζℓ). By Proposition 8.4we have (Cl(Kz)/Cl(Kz)

ℓ)[τ + 1] = {1}, so that by Lemma 2.6 we have Sℓ(Kz)[T∗] ≃ (U(Kz)/U(Kz)

ℓ)[τ −g(ℓ−1)/2]. By Theorem 2.3 of [10] we deduce that Sℓ(Kz)[T

∗] is trivial if ℓ ≡ 3 (mod 4), ℓ 6= 3, and when

ℓ ≡ 1 (mod 4) that it is an Fℓ-vector space of dimension 1. If ε is a fundamental unit of K = Q(√ℓ), then

since τ acts on ε as Galois conjugation of K/Q, we have ετ(ε) = NK/Q(ε) = ±1, which is an ℓth power. It

follows that Sℓ(Kz)[T∗] = {εj , j ∈ Fℓ}.

The sizes of the ray Selmer groups are then established by Lemma 8.6. �

Remark 8.8. The assumption that ℓ 6= 3 is required when applying Theorem 2.3 of [10], and indeed (2) ofthe proposition is false for ℓ = 3 (see Proposition 7.3 of [14]).

This proposition, in combination with Corollary 8.3, gives the size of |Gb| in the special case, with possibleexceptions ℓ ≡ 1 (mod 4) larger than 2 · 1011. In the general case we have the following:

Corollary 8.9. [15, Corollary 3.5] Assume that we are in the general case.

(1) We have a canonical isomorphism Gb ≃ Hom(Sb∗∩K(K),µℓ).(2) In particular

|Gb| = ℓr(b) with r(b) = 1− r2(D)− z(b) + rkℓ(Clb∗∩K(K)) ,

with z(b) = (2, 1, 0, 0) respectively, the second case occurring only if ℓ|D.(3) In particular still, if D < 0 and ℓ ∤ h(D) then Gb is trivial for all b ∈ B.

Proof. (1) is immediate. Lemma 2.6, and Proposition 2.12 of [10], the proofs of which adapt to K withoutchange, yield

|Sb1(K)||Zb1/Zℓb1| = ℓ1−r2(D)|Clb1(K)/Clb1(K)ℓ| ,

where Zb1 = (ZK/b1)∗ and b1 = b∗ ∩ K. This gives (2) with z(b) = dimFℓ

(Zb1/Zℓb1), and to finish we

compute for b as in (7.1):

• If ℓ ∤ D, we have b∗ ∩K = (ℓ2ZK , ∗, ℓZK ,ZK).• If ℓ | D with ℓZK = p2ℓ , we have b∗ ∩K = (p3ℓ , p

2ℓ , pℓ,ZK).

�

Note that (3) is a generalization of Proposition 7.7 of [14].Since the triviality of Gb for all b is equivalent to the vanishing of the “remainder term” φD(s) of Corollary

7.4, we conclude that Φℓ(K, s) is given by a single Euler product in a wide class of examples:


Corollary 8.10. Assume that ℓ ≥ 5, D < 0, and that either we are in the special case (so that ℓ ≡ 3(mod 4)), or that we are in the general case with ℓ ∤ h(D). Then we have

∑

L∈Fℓ(K)

1

f(L)s= − 1

ℓ− 1+

1

ℓ− 1Lℓ(s)

∏


(1 +

ℓ− 1

ps

),

where Lℓ(s) is as above.

Note that for ℓ = 3, which we have excluded here, the possible nontriviality of re(b) forces us to alsodistinguish between D ≡ 3 and D ≡ 6 (mod 9).

Examples with ℓ = 5:

∑

L∈F5(Q(√−1))

1

f(L)s= −1

4+

1

4

(1 +

4

52s

) ∏

p≡±1 (mod 20)

(1 +

4

ps

).

∑

L∈F5(Q(√−15))

1

f(L)s= −1

4+

1

4

(1 +

4

5s

) ∏

p≡±1 (mod 30)

(1 +

4

ps

).

9. Transformation of the Main Theorem

We now prove, as we did in [16] for the case of ℓ = 3, that the characters of Gb appearing in Theorem 6.1can be given a simpler description, in terms of the splitting of primes in degree ℓ extensions of k. Our mainresult along these lines extends Theorem 4.1 of [16] and Proposition 3.7 of [15], and does not assume thatk = Q, and thus is new even for ℓ = 3.

For the case k = Q we will further specialize the result and obtain an explicit formula, relying (in thegeneral case) on the results of [15]. We will assume that we are in either the general case or in the specialcase with ℓ ≡ 1 (mod 4). Recall that in the special case with k = Q, ℓ ≡ 3 (mod 4), and ℓ > 3, Gb istrivial and Corollary 8.10 already gives a simple description of Φℓ(K, s). For simplicity’s sake we will omitthe special case with k 6= Q, ℓ ≡ 3 (mod 4); as we will see below the group theory would work out a bitdifferently.

Recall that the Frobenius group Fℓ = Cℓ ⋊ Cℓ−1 is the non-abelian group of order ℓ(ℓ− 1) given as

〈τ, σ : τ ℓ−1 = σℓ = 1, τστ−1 = σh〉 ,for any primitive root h (mod ℓ). As may be easily checked, Cℓ−1 is not normal in Fℓ, nor is any nontrivialsubgroup of Cℓ−1; moreover, there are ℓ subgroups isomorphic to Cℓ−1, generated by τσi for 0 ≤ i ≤ ℓ− 1,and all of these subgroups are conjugate. We will say that a degree ℓ field extension E/k is an Fℓ-extensionif its Galois closure has Galois group Fℓ over k.

Now, let K,Kz , τ, τ2 be defined as before. In the general case recall that K ′ was defined to be the mirrorfield of K, e.g., the subfield of Kz fixed by τ (ℓ−1)/2τ2; in the special case write K ′ = Kz = kz. We choseτ ∈ Gal(kz/k) and a primitive root g (mod ℓ) with τ(ζℓ) = ζgℓ . In the general case τ lifts uniquely to anelement of Gal(Kz/K) and restricts to a unique element of Gal(K ′/k), so in either case the choice of g(mod ℓ) uniquely determines τ ∈ Gal(K ′/k).

Theorem 9.1. Assume, if ℓ ≡ 3 (mod 4), that we are in the general case. For each b ∈ B (as in Theorem6.1), there exists a bijection between the following sets:

• Characters χ ∈ Gb, up to the equivalence relation χ ∼ χa for each a coprime to ℓ.• Subgroups of index ℓ of Gb.


• Fℓ-extensions E/k (up to isomorphism), whose Galois closure E′ contains K ′ and whose conductorf(E′/K ′) divides b′ = b ∩K, and such that τστ−1 = σg for τ ∈ Gal(K ′/k) as described above andany generator σ of Gal(E′/K ′).

Moreover, for each corresponding pair (χ,E) and each prime p ∈ D ∪ Dℓ, the following is true: we havep ∈ D′(χ) ∪ D′

ℓ(χ) if and only if p is totally split in E; equivalently, p 6∈ D′(χ) ∪ D′ℓ(χ) if and only if p is

totally inert or totally ramified in E.

Proof. The proof borrows heavily from those of Proposition 4.1 of [16] and Proposition 3.7 of [15].The correspondence between the first two sets is immediate: Gb is elementary ℓ-abelian, and characters

correspond to their kernels.

By Proposition 6.3, regard Gb asClb′(K

′)

Clb′(K′)ℓ[τ ∓ g], where the sign is − in the general case and + in the

special case. If we setG′b = Clb′(K

′)/Clb′(K ′)ℓ, by Lemmas 2.1 and 2.3 we have the orthogonal decompositionG′

b = Gb×G′′b , where G

′′b is the direct sum of all of the other eigenspaces for the actions of τ . Thus, subgroups

of Gb of index ℓ correspond to subgroups B of Clb′(K′) of index ℓ containing G′′

b .By class field theory, there exists a unique abelian extension E′/K ′, with Galois group Cℓ and conductor

dividing b′, for which the Artin map induces an isomorphism Clb′(K′)/B ≃ Gal(E′/K ′). The uniqueness

forces E′ to be Galois over k; here we use that b′, B, and Clb′(K′) are preserved by Gal(K ′/k). For each

fixed b, we obtain a different E′ for each B.Because the action of Gal(K ′/k) on Clb′(K

′)/Bχ matches its conjugation action on Gal(E′/K ′), we have

(9.1) Gal(E′/k) = 〈τ, σ : τ ℓ−1 = σℓ = 1, τστ−1 = σ±g〉 ≃ Fℓ ,

and we take E to be the fixed field of 〈τ〉 (or, alternatively, of any conjugate subgroup). Note that −g is nota primitive root if ℓ ≡ 3 (mod 4), so that in the special case with ℓ ≡ 3 (mod 4) the group (9.1) contains

τ (ℓ−1)/2 in its center and is not isomorphic to Fℓ.

It must finally be proved that whether p ∈ D′(χ) or not is determined by its splitting in E. Proposition2.15 or Corollary 2.12 implies that D ∪ Dℓ is precisely the set of primes p which split completely in K ′/k,and by definition D′(χ)∪D′

ℓ(χ) is the set of primes p ∈ D∪Dℓ for which one (equivalently, all) of the primespK ′ of K ′ above p split completely in E′. If pK ′ splits completely in E′, then so does p, so p also splitscompletely in E/k. Conversely, if any pK ′ is completely ramified or inert in Kz, then p must also do thesame in each E, since ramification and inertial degrees are multiplicative and [E′ : E] = ℓ− 1. �

For ℓ = 3 and k = Q in the general case, in [16] we further applied a theorem of Nakagawa to give aprecise description of all the extensions E/Q occurring in the statement of Theorem 9.1 in terms of theirdiscriminants. Using this, we obtained the formula

(9.2)2

3r2(D)Φ3(Q(

√D), s) = M1(s)

∏(

−3Dp

)=1

(1 +

2

ps

)+

∑

E∈L3(D)

M2,E(s)∏

(−3Dp

)=1

(1 +

ωE(p)

ps

),

where: L3(D) is the set of all cubic fields of discriminant −D/3, −3D, and −27D; ωE(p) is 2 or −1 dependingon whether p is split or inert in E, as in Theorem 9.1; and M1(s) and M2,E(s) are 3-adic factors (a sum ofthe appropriate Ab(s)).

9.1. Explicit computations for k = Q in the special case. For ℓ = 3, we have the following explicitformula (corresponding to pure cubic fields), which was previously proved in [14].

∑

L∈F3(Q(√−3))

1

f(L)s= −1

2+

1

6

(1 +

2

3s+

6

32s

)∏

p 6=3

(1 +

2

ps

)+

1

3

∏

p

(1 +

ωE(p)

ps

),


where E is the cyclic cubic field defined by x3 − 3x− 1 = 0 of discriminant 34, and

ωE(p) =

{−1 if p is inert or totally ramified in E ,

2 if p is totally split in E (equivalently, p ≡ ±1 (mod 9)) .

For ℓ ≡ 3 (mod 4) and ℓ > 3, a generalization was proved in Corollary 8.10. For ℓ ≡ 1 (mod 4), thegeneralization is more complicated due to the nontriviality of Gb. Define a polynomial

P (x) = −2iTℓ(ix/2) =

(ℓ−1)/2∑

k=0

ℓ(ℓ− k − 1)!

k!(ℓ− 2k)!xℓ−2k .

Here Tℓ(x) is the Chebyshev polynomial of the first kind, satisfying

(9.3) P (x− x−1) = xℓ − x−ℓ ,

so that x−1P (x) is the minimal polynomial of ζℓ − ζ−1ℓ .

Proposition 9.2. Assume that ℓ ≡ 1 (mod 4) satisfies the Ankeny-Artin-Chowla conjecture, and let ε be a

fundamental unit of Q(√ℓ). Then we have

∑

L∈Fℓ(Q(√ℓ))

1

f(L)s= − 1

ℓ− 1+

1

ℓ(ℓ− 1)

(1 +

ℓ− 1

ℓs

) ∏

p≡1 (mod ℓ)

(1 +

ℓ− 1

ps

)+

1

ℓ

∏

p

(1 +

ωE(p)

ps

),

where E is the Fℓ-field defined by P (x)− Tr(ε) = 0 of discriminant ℓ(3ℓ−1)/2, and

ωE(p) =

−1 if p is inert or totally ramified in E ,

ℓ− 1 if p is totally split in E ,

0 otherwise.

If ℓ ≡ 1 (mod 4) does not satisfy the Ankeny-Artin-Chowla conjecture, we have the same formula, but withDisc(E) = ℓℓ−2 and ωE(ℓ) = ℓ− 1.

Corollary 9.3. Let ℓ ≡ 1 mod 4. Then there exist Dℓ-fields ramified only at ℓ if and only if the Ankeny-Artin-Chowla conjecture is false for ℓ.

Proof. Immediate; for ℓ not satisfying the conjecture, the field is unique and has discriminant ℓ3(ℓ−1)

2 . �

Remark 9.4. This corollary recovers and strengthens a result of Jensen and Yui [26, Theorem I.2.2], whoproved that if ℓ ≡ 1 (mod 4) is regular, then there are no Dℓ-fields with discriminant a power of ℓ. (Thiscan also be seen for ℓ ≡ 3 (mod 4) from Corollary 8.10.)

The connection to the Ankeny-Artin-Chowla conjecture was previously observed by Lemmermeyer [31],who suggested that a proof of Corollary 9.3 may exist somewhere in the literature.

Before beginning the proof of Proposition 9.2 we establish the following:

Lemma 9.5. We have

Disc(Nz) =

{ℓ(3ℓ

2−2ℓ−3)/2 if AAC is true,

ℓℓ(ℓ−2) if AAC is false.

In addition, in the extension Nz/Qz the prime ideal (1− ζ)ZQz is totally ramified if AAC is true and totallysplit otherwise.


Proof. The field Nz is a Kummer extension of Kz = Qz with defining equation xℓ − ε = 0, so that

Disc(Nz) = ±NQz/Q(d(Nz/Qz))Disc(Qz)ℓ = ±ℓℓ(ℓ−2)NQz/Q(f(Nz/Qz))

ℓ−1 ,

where f(Nz/Qz) is the conductor.By [10, Theorem 3.7] applied to K = Qz and α = ε which is a unit, we have f(Nz/Qz) = (1 − ζ)ℓ+1−Aε ,

where Aε = ℓ + 1 if xℓ ≡ ε (mod (1 − ζ)ℓ) has a solution in Qz, and otherwise Aε is the maximal k suchthat xℓ ≡ ε (mod (1 − ζ)k) has a solution. By Lemma 8.6 we have Aε = (ℓ − 1)/2 (resp., Aε = ℓ + 1) if

AAC is true (resp., false), hence f(Nz/Qz) = (1 − ζ)(ℓ+3)/2ZQz (resp., f(Nz/Qz) = ZQz), from which theformula follows (note that the sign of the discriminant is positive since Qz hence Nz is totally complex). Inaddition, if AAC is false, so that Ck is soluble for all k, then Hecke’s Theorem [8, 10.2.9] (an extension of[10, Theorem 3.7]) implies that (1−ζ)ZQz is totally split, while if AAC is true then it is totally ramified. �

Proof of Proposition 9.2. The result follows for an undetermined E by Theorem 9.1 and Proposition 8.7. Todetermine E, observe that Proposition 8.1 and the proof of Proposition 8.7 imply that Nz = Kz(ǫ

1/ℓ), andthat the considerations in the proof of Theorem 9.1 allow us to take E to be any of the (conjugate) degreeℓ subfields of Nz, so that it suffices to exhibit one.

We take E = Q(ε1/ℓ − ε−1/ℓ) for any fixed choice of ε1/ℓ, recalling that the fundamental unit has norm−1. Then the minimal polynomial of E is P (x)− Tr(ε) by construction, or more precisely by (9.3).

It remains only to argue that

Disc(E) =

{ℓ(3ℓ−1)/2 if AAC is true,

ℓℓ−2 if AAC is false.

We assume that AAC is true (if false, a similar proof applies). On the one hand we have

Disc(Nz) = Disc(E)ℓ−1NE/Q(d(Nz/E)) ,

in other words taking valuations and using the proposition:

(ℓ−1)vℓ(Disc(E)) = (3ℓ2−2ℓ−3)/2−vℓ(NE/Q(d(Nz/E))) = (ℓ−1)(3ℓ−1)/2+ ℓ−2−vℓ(NE/Q(d(Nz/E))) .

On the other hand, the extension Nz/E is of degree ℓ − 1 hence tame, so vℓ(NE/Q(d(Nz/E))) ≤ ℓ − 2.Divisibility by ℓ − 1 thus implies the result, together with the additional result that NE/Q(d(Nz/E)) =

ℓℓ−2 = Disc(Qz). �

We make some additional observations concerning Proposition 9.2:

Remarks 9.6. (1) In the equation for E we may replace Tr(ε) by Tr(±εm) for any odd m ∈ Z coprimeto ℓ.

(2) Assuming AAC, the last product may be written as(1− 1

ℓs

) ∏

p≡1 (mod ℓ)

(1 +

ωE(p)

ps

).

(3) When p ≡ 1 (mod ℓ) then p is totally split in E if and only if ε(p−1)/ℓ ≡ 1 (mod p), and otherwisep is inert in E.

(4) If in addition Qz = Q(ζℓ) has class number 1, then p is totally split in E if and only if p = NQz/Q(π)for some π ≡ 1 (mod ℓ) in Qz.

For (1), it is easily seen that our construction still produces a degree ℓ subfield E. (2) follows because ℓis totally ramified in E.

To prove (3), again apply [8, Theorem 10.2.9]: p is totally split in E iff it is in Nz/Qz, hence iff xℓ ≡ ε(mod p) is soluble in Qz. (Here p is any prime of Qz above p, which must have degree 1 since p ≡ 1 (mod ℓ)


is totally split in Qz.) This is equivalent to ε(p−1)/ℓ ≡ 1 (mod p), which by Galois theory will then be truefor all primes p above p since for any σ ∈ Gal(Qz/Q) we have either σ(ε) = ε or σ(ε) = −ε−1, and (p− 1)/ℓ

is even so the sign disappears. Hence this is equivalent to the condition ε(p−1)/ℓ ≡ 1 (mod p), as desired.Finally, (4) follows from Eisenstein’s reciprocity law.

9.2. Explicit computations for k = Q in the general case. Let k = Q. In Theorem 9.1 we saw thatcharacters χ of Gb (up to the equivalence χ ∼ χa for (a, ℓ) = 1) correspond to degree ℓ fields E havingcertain properties. In our companion paper [15] with Rubinstein-Salzedo, we further proved the following:

Theorem 9.7. [15] Suppose that k = Q and K = Q(√D) with D 6= 1,±ℓ, so that we are in the general

case, and as before let K ′ be the mirror field of K.Then the fields E enumerated in Theorem 9.1 are precisely those Fℓ-fields E whose Galois closure contains

K ′, subject to the condition τστ−1 = σg described there, satisfying the following additional conditions:

• E is totally real if D < 0, and has ℓ−12 pairs of complex embeddings if D > 0.

• |Disc(E)| has the form ℓk+bDℓ−12 , where k and b satisfy

(9.4)

k ∈ {0, 2}, b = ℓ− 2 if ℓ ∤ D ,

k ∈ {0, (ℓ+ 3)/2} , b = ℓ−32 if ℓ | D and ℓ ≡ 1 (mod 4) ,

k ∈ {0, (ℓ+ 5)/2} , b = ℓ−52 if ℓ | D and ℓ ≡ 3 (mod 4) .

Moreover, if E is any Fℓ-field satisfying these last two properties, then its Galois closure automaticallycontains K ′.

Recall that we have b ∈ B = {1, (ℓ)1/2, (ℓ), (ℓ)ℓ/(ℓ−1)}, with the possibility (ℓ)1/2 occurring only if ℓ|D.

The complete list of fields enumerated in Theorem 9.7 corresponds to b = (ℓ)ℓ/(ℓ−1). A careful reading ofthe proof of Theorem 9.7 (in [15]), with k having the same meaning above as in [15, Section 4], shows thatthe remaining b correspond to the following possibilities for k in (9.4):

Condition on D b = 1 b = (ℓ)1/2 b = (ℓ) b = (ℓ)ℓ/(ℓ−1)

ℓ ∤ D k = 0 - k = 0, 2 k = 0, 2ℓ | D and ℓ ≡ 1 (mod 4) k = 0 k = 0 k = 0, (ℓ + 3)/2 k = 0, (ℓ+ 3)/2ℓ | D and ℓ ≡ 3 (mod 4) k = 0 k = 0 k = 0, (ℓ + 5)/2 k = 0, (ℓ+ 5)/2

One exception occurs for ℓ = 3: Only k = 0 corresponds to b = (ℓ) when ℓ | D; this is because theinequality (ℓ + 5)/2 ≤ ℓ − 1 is true for all ℓ ≡ 3 (mod 4) except for ℓ = 3. (Note also for ℓ = 3 that thisresult is equivalent to part of Proposition 4.1 in [16].)

This is sufficient to obtain an explicit formula for Φℓ(K, s) for any K and ℓ, provided that the appropriateFℓ-fields can be tabulated. We present two examples, which we also double-checked numerically using aprogram written in PARI/GP [39].

Example 9.8. Let K = Q(√13) and ℓ = 5. Then, we have

∑

L∈F5(Q(√13))

1

f(L)s= −1

4+

1

20

(1 +

4

25s

)∏

p

(1 +

4

ps

)+

1

5

(1− 1

25s

)∏

p

(1 +

ωE(p)

ps

)

= 59−s + 409−s + 475−s + 619−s + 709−s + 1009−s + · · ·+ 4 · 24131−s + · · · ,where the products are over primes p ≡ 1, 16, 19, 24, 34, 36, 44, 51, 54, 56, 59, 61 (mod 65), E is the field de-fined by the polynomial x5 + 5x3 + 5x− 3, and 24131 = 59 · 409.


Example 9.9. Let K = Q(√−7 · 41) and ℓ = 7. Then, we have

∑

L∈F7(Q(√−287))

1

f(L)s= −1

6+

1

6

(1 +

6

7s

)∏

p

(1 +

6

ps

)+

(1− 1

7s

)∏

p

(1 +

ωE(p)

ps

)

= 1 + 7 · 301−s + 7 · 337−s + 7 · 581−s + 7 · 791−s + · · ·+ 42 · 296897−s + · · · ,

where the products are over primes p ≡(Dp

)(mod ℓ) excluding p = ℓ, E is the field defined by the polynomial

x7 − 14x5 + 56x3 − 56x− 15, and 296897 = 337 · 881.

10. Upper bounds for counting dihedral extensions of Q

Write Nℓ(Dℓ,X) for the number of Dℓ-fields L with |Disc(L)| < X. Kluners [29] proved that N(Dℓ,X) ≪X

3ℓ−1

+ǫ and in this section we prove Theorem 1.1, obtaining N(Dℓ,X) ≪ X3

ℓ−1− 1

ℓ(ℓ−1)+ǫ

as an easy conse-quence of work of Ellenberg, Pierce, and Wood [22].

As in the introduction, for each Dℓ-field L we have |Disc(L)| = nℓ−1|D| ℓ−12 , where n ∈ Z and Q(

√D)

is the quadratic resolvent field of D. For each D and n, the multiplicity of L with this discriminant isOℓ(ℓ

rkℓ(Cl(D))+2ω(n)); this follows from [29, Lemma 2.3] or alternatively Theorem 7.3 here.We therefore have that

N(Dℓ, x) ≪∑

|D|<X2/(ℓ−1)

Cl(D)[ℓ]∑

n<(X/D(ℓ−1)/2)1/(ℓ−1)

ℓ2ω(n)

≪ X1

ℓ−1+ǫ

∑

|D|<X2/(ℓ−1)

D− 12Cl(D)[ℓ].

By [22, Theorem 1.1], for all but O(X2

ℓ−1·(1− 1

2ℓ )) discriminants D in the sum, we have Cl(D) ≪ D12− 1

2ℓ+ǫ,

and the contribution of these D is

≪ X1

ℓ−1+2ǫ

∑

D<X2/(ℓ−1)

D− 12+ 1

2− 1

2ℓ ≪ X1

ℓ−1+2ǫX

2ℓ−1

·(1− 12ℓ) = X

3ℓ−1

− 1ℓ(ℓ−1)

+2ǫ.

By the trivial bound on Cl(D), the contribution of the remaining D is also ≪ X1

ℓ−1+2ǫ · X 2

ℓ−1·(1− 1

2ℓ),completing the proof.

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https://arxiv.org/abs/1606.06103

http://mathoverflow.net/questions/13152/dihedral-extensions-and-the-ankeny-artin-chowla-conjecture

http://scholarcommons.sc.edu/cgi/viewcontent.cgi?article=3870&context=etd

http://pari.math.u-bordeaux.fr/


Universite de Bordeaux, Institut de Mathematiques, U.M.R. 5251 du C.N.R.S, 351 Cours de la Liberation,

33405 TALENCE Cedex, FRANCE

E-mail address: [email protected]

Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, SC 29208, USA

E-mail address: [email protected]

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